Product Representation: Transition between different representations of a function

We will begin by using the distributive property in order to expand the following expression.

(a+1)⋆(b+2) = ab+2a+b+2

We will then proceed to insert the known values into the equation and solve as follows:

(x-2)(x+5) =

x²-2x+5x+-2*5=

x²+3x-10

And that's the solution!

To find the standard representation of the quadratic function $f(x) = (x - 6)(x - 2)$, follow these steps:

• Step 1: Apply the FOIL method to expand the expression.
  - First: Multiply the first terms: $x \times x = x^2$.
  - Outside: Multiply the outer terms: $x \times (-2) = -2x$.
  - Inside: Multiply the inner terms: $(-6) \times x = -6x$.
  - Last: Multiply the last terms: $(-6) \times (-2) = 12$.
• Step 2: Combine the results from the FOIL method.
  - Combine all the expanded terms: $x^2 - 2x - 6x + 12$.
• Step 3: Simplify by combining like terms.
  - Combine the $x$-terms: $-2x - 6x = -8x$.
  - The expanded and simplified form is: $f(x) = x^2 - 8x + 12$.

By expanding and simplifying the given product, we have converted it to its standard form. Therefore, the standard representation of the function is $f(x) = x^2 - 8x + 12$.

The correct choice from the provided options is choice 2: $f(x) = x^2 - 8x + 12$.

To find the standard representation of the quadratic function given by $f(x) = (x+2)(x-4)$, we will expand the expression using the distributive property, commonly known as the FOIL method (First, Outer, Inner, Last):

• First: Multiply the first terms in each binomial: $x \cdot x = x^2$.
• Outer: Multiply the outer terms in the binomials: $x \cdot (-4) = -4x$.
• Inner: Multiply the inner terms: $2 \cdot x = 2x$.
• Last: Multiply the last terms in each binomial: $2 \cdot (-4) = -8$.

Now, let's combine these results:

The expression becomes $x^2 - 4x + 2x - 8$.

Next, we combine like terms:

The terms involving $x$ are $-4x + 2x$, which simplifies to $-2x$.

Thus, the expression simplifies to: $f(x) = x^2 - 2x - 8$

Upon comparing this result to the provided choices, we find that it matches choice 3.

Therefore, the standard representation of the function is $f(x) = x^2 - 2x - 8$.

To find the standard representation of the quadratic function $f(x) = 3x(x + 4)$, follow these steps:

• Step 1: Apply the distributive property to expand the expression $x(x + 4)$.
  Using this property, we have:
  $x(x + 4) = x \cdot x + x \cdot 4 = x^2 + 4x$.
• Step 2: Multiply each term by the coefficient outside the parenthesis, which is 3.
  This gives us:
  $3(x^2 + 4x) = 3 \cdot x^2 + 3 \cdot 4x$.
• Step 3: Simplify by performing the multiplication.
  $3x^2 + 12x$.

Therefore, the standard representation of the function is $f(x) = 3x^2 + 12x$. This matches choice 3 in the provided answers.

To solve this problem, we'll convert the given function from its factored form to the standard form using the distributive property. The given function is $f(x) = -x(x - 8)$.

Let's go through the necessary steps:

• Step 1: Apply the distributive property to expand the expression.
  $f(x) = -x(x - 8) = -x \cdot x + (-x) \cdot (-8)$
• Step 2: Simplify each term.
  $-x \cdot x = -x^2$ and $(-x) \cdot (-8) = +8x$.
• Step 3: Combine the terms to express $f(x)$ in standard form:
  $f(x) = -x^2 + 8x$.

Therefore, the standard representation of the function is $f(x) = -x^2 + 8x$.

Comparing this result to the multiple-choice options, we can see that the correct choice is option 3: $f(x)=-x^2+8x$.

To solve this problem and find the standard representation of the function $f(x) = (x+1)(x-1)$, we will expand the product using the distributive property, often recalled as FOIL (First, Outer, Inner, Last) for the product of two binomials.

Let's proceed step-by-step:

• Step 1: Apply the distributive property:
  $f(x) = (x+1)(x-1)$ would become:
• First terms: $x \cdot x = x^2$
• Outer terms: $x \cdot (-1) = -x$
• Inner terms: $1 \cdot x = x$
• Last terms: $1 \cdot (-1) = -1$

Step 2: Combine all the terms obtained from the FOIL method:
$x^2 - x + x - 1$

Step 3: Simplify the expression by combining like terms:
The terms $-x$ and $x$ cancel each other out, simplifying to:
$f(x) = x^2 - 1$

Thus, the standard representation of the function is $f(x) = x^2 - 1$.

To find the standard representation of the function $f(x) = (2x + 1)(x - 2)$, we'll follow these steps to expand and simplify the expression:

• Step 1: Distribute each term of the first binomial over each term of the second binomial using the FOIL method.
• Step 2: Combine like terms to express the function in standard quadratic form.

Now, let's expand the expression:
1. Multiply the first terms: $2x \cdot x = 2x^2$
2. Multiply the outer terms: $2x \cdot (-2) = -4x$
3. Multiply the inner terms: $1 \cdot x = x$
4. Multiply the last terms: $1 \cdot (-2) = -2$

Next, we combine these results:
- The $2x^2$ term remains as is.
- Add the linear terms: $-4x + x = -3x$
- The constant term is $-2$.

Thus, the expanded and simplified form of the function is:
$f(x) = 2x^2 - 3x - 2$

The final expression in standard form is $f(x) = 2x^2 - 3x - 2$.

To find the standard form of the given quadratic function $f(x) = (x+3)(-x-4)$, we will expand it using the distributive property.

Step 1: Expand the product.
Using the distributive property (or FOIL method):

$f(x) = (x+3)(-x-4)$

Apply distribution:
First: $x \cdot -x = -x^2$
Outside: $x \cdot -4 = -4x$
Inside: $3 \cdot -x = -3x$
Last: $3 \cdot -4 = -12$

Step 2: Combine all terms together:

$f(x) = -x^2 - 4x - 3x - 12$

Step 3: Simplify by combining like terms:
Combine the $x$ terms:

$f(x) = -x^2 - 7x - 12$

Therefore, the standard representation of the function is $f(x) = -x^2 - 7x - 12$.

The correct choice from the given options is choice 4.

$f(x)=-x^2-7x-12$

To solve the problem of finding the product (factored) representation of the quadratic function $f(x) = x^2 - 7x + 12$, we proceed as follows:

• Step 1: Identify the function, which is $f(x) = x^2 - 7x + 12$.
• Step 2: We need to factor this quadratic expression. We're looking for two numbers whose product is 12 and whose sum is -7.
• Step 3: The factor pairs of 12 are $(1, 12)$, $(2, 6)$, $(3, 4)$, including negative pairs because the sum must be negative.
• Step 4: Consider the pair $(-3, -4)$. The product $(-3) \times (-4)$ equals 12, and the sum $(-3) + (-4)$ equals -7.

Therefore, the factors of the quadratic expression are $x - 3$ and $x - 4$. This implies that the function $f(x)$ can be expressed in product form as:

$f(x) = (x - 3)(x - 4)$

This means the correct factorization is $(x - 3)(x - 4)$, which corresponds to choice 3 from the given options.

Thus, the representation of the product of the function is $(x - 3)(x - 4)$.

The problem requires finding the product representation of the quadratic function $f(x) = x^2 - 2x - 3$.

Let's execute the factorization of the quadratic equation:

• The standard form for the function is $f(x) = ax^2 + bx + c$. Here, $a = 1$, $b = -2$, $c = -3$.
• We seek two numbers that multiply to $c = -3$ and sum to $b = -2$.
• Checking possible integer pairs: $(-3, 1)$ can accomplish this, since $-3 \times 1 = -3$ and $-3 + 1 = -2$.
• The factorization becomes $f(x) = (x - 3)(x + 1)$.

To verify, we can expand the binomials:

$(x - 3)(x + 1) = x^2 + x - 3x - 3 = x^2 - 2x - 3$.

This matches the original polynomial, confirming the product representation is correct.

In conclusion, the factorization or product representation of the given quadratic function is $\mathbf{(x-3)(x+1)}$.

To solve the problem of factoring the quadratic expression $f(x) = x^2 - 3x - 18$, we will use the following method:

• Step 1: Identify and understand the quadratic expression, which is given in standard form: $ax^2 + bx + c$. For this expression, $a = 1$, $b = -3$, and $c = -18$.
• Step 2: Compute the product of $a$ and $c$, which yields $1 \cdot (-18) = -18$. We need to find two numbers whose product is $-18$ and whose sum is $-3$.
• Step 3: Look for pairs of factors of $-18$: - $1, -18$ - $-1, 18$ - $2, -9$ - $-2, 9$ - $3, -6$ - $-3, 6$

Among these, the pair $(3, -6)$ adds up to $-3$ and multiplies to $-18$.

• Step 4: Rewrite the quadratic expression using these numbers to represent the middle term:
  $x^2 - 3x - 18 = x^2 + 3x - 6x - 18$.
• Step 5: Group the terms to facilitate factoring:
  $(x^2 + 3x) + (-6x - 18)$.
• Step 6: Factor out the common factors in each grouped terms:
  $x(x + 3) - 6(x + 3)$.
• Step 7: Factor out the common binomial:
  $(x - 6)(x + 3)$.

Therefore, the factorized form of the quadratic function $f(x) = x^2 - 3x - 18$ is $(x - 6)(x + 3)$.

To determine the product representation of $f(x) = x^2 + x - 2$, we can factor the quadratic equation by following these steps:

• Step 1: Identify the product $ac = 1 \times (-2) = -2$ and sum $b = 1$.
• Step 2: Find two numbers that multiply to $-2$ and add to $1$. These numbers are $2$ and $-1$.
• Step 3: Rewrite the middle term using these numbers: $x^2 + 2x - 1x - 2$.
• Step 4: Factor by grouping:
  - Group $x^2 + 2x$ and $-1x - 2$ as separate pairs:
  - $x(x + 2) - 1(x + 2)$.
• Step 5: Factor out the common terms:
  $(x + 2)(x - 1)$.

Thus, the product representation of the function is $(x + 2)(x - 1)$.

To solve this problem, we'll follow these steps:

• Step 1: Use the FOIL method to expand the product of binomials.
• Step 2: Combine like terms to simplify the expression.

Now, let's work through each step:

Step 1: Expand the product $(-x+2)(x+3)$ using the FOIL method:
First terms: $-x \cdot x = -x^2$
Outer terms: $-x \cdot 3 = -3x$
Inner terms: $2 \cdot x = 2x$
Last terms: $2 \cdot 3 = 6$

This gives us the expression:
$-x^2 - 3x + 2x + 6$

Step 2: Combine like terms:
Combine $-3x$ and $2x$ to get $-x$.
Thus, the expression simplifies to:
$-x^2 - x + 6$

Therefore, the standard form of the function is $f(x) = -x^2 - x + 6$.

To solve this problem, we need to convert the given function $f(x) = -(x+1)(x-1)$ from its current product form to standard form.

Let's follow these steps:

• Step 1: Recognize that the expression $(x+1)(x-1)$ is a standard difference of squares formula, expressed as $(a+b)(a-b) = a^2 - b^2$, where $a = x$ and $b = 1$.
• Step 2: According to the formula, $(x+1)(x-1) = x^2 - 1^2 = x^2 - 1$.
• Step 3: Substitute this result back into the function: $f(x) = -(x^2 - 1)$.
• Step 4: Simplify the expression by distributing the negative sign: $-(x^2 - 1) = -x^2 + 1$.

Therefore, the function in its standard form is $f(x) = -x^2 + 1$.

This matches with choice 1: $f(x)=-x^2+1$.

To solve the problem of factoring the quadratic expression $f(x) = x^2 + 12x + 32$, we will follow these steps:

• Identify the pairs of integers whose product is 32.
• Find the pair that also adds up to 12.
• Use this pair to express the quadratic in its factored form.

Let's proceed with these steps:

Step 1: List all pairs of integers that multiply to 32:
1 and 32
2 and 16
4 and 8

Step 2: Determine which pair of these adds up to 12:
Checking each:
$1 + 32 = 33$
$2 + 16 = 18$
$4 + 8 = 12$

The pair 4 and 8 adds up to 12.

Step 3: Use this pair to factor the quadratic expression:
Thus, $f(x) = (x + 4)(x + 8)$.

Therefore, the factored form of $f(x) = x^2 + 12x + 32$ is $(x + 8)(x + 4)$.

To find the product representation of the function $f(x) = x^2 - 16x + 64$, we expect it to be a perfect square trinomial.

First, recognize that the given quadratic form is $a^2 - 2ab + b^2$. Comparing it with $x^2 - 16x + 64$:

• The first term $x^2$ suggests $a = x$.
• The middle term $-16x$ can be interpreted as $-2ab$. So, $-2a = -16$, solving gives $2b = 16$, hence $b = 8$.
• The last term $(64)$ is $b^2$, confirms that $8^2 = 64$.

This means our expression is:

• $f(x) = (x - 8)^2$

Thus, the product form or factored representation of the function is $(x-8)^2$.

The final answer is: $(x-8)^2$.

To solve this problem, we will factor the quadratic function $f(x) = x^2 - 5x - 50$ into two binomials:

• Step 1: Identify the values of $a = 1$, $b = -5$, and $c = -50$ in the quadratic expression $ax^2 + bx + c$.
• Step 2: We look for two numbers that multiply to $c = -50$ and add up to $b = -5$.
• Step 3: Consider the factor pairs of $-50$. Possible pairs include $(-1, 50)$, $(1, -50)$, $(-2, 25)$, $(2, -25)$, $(-5, 10)$, and $(5, -10)$.
• Step 4: The correct pair is $(5, -10)$ because $5 \times (-10) = -50$ and $5 + (-10) = -5$.
• Step 5: Express $f(x)$ in its factored form using these numbers: $f(x) = (x + 5)(x - 10)$.
• Step 6: Verify the factorization by expanding: $(x + 5)(x - 10) = x^2 - 10x + 5x - 50$, which simplifies to $x^2 - 5x - 50$, confirming correctness.

Therefore, the correct factorization of the quadratic $f(x) = x^2 - 5x - 50$ is $(x + 5)(x - 10)$.

Thus, the product representation of the function is $(x+5)(x-10)$.

To solve this problem, we need to express the quadratic function $f(x) = x^2 - 6x + 9$ as a product of binomials.

First, observe whether the expression can be written as a perfect square trinomial. It helps to compare it with the standard perfect square form: $(x-a)^2 = x^2 - 2ax + a^2$.

In our quadratic, we have:

• The quadratic term $x^2$ matches exactly.
• The linear term $-6x$ suggests $-2a = -6$. Solving for $a$, we have $a = 3$. Hence, the expression of vertex form should be $(x-3)^2$.
• The constant term is $9$. In a perfect square trinomial, this term would also be $3^2 = 9$, which fits perfectly.

Thus, the original quadratic function $x^2 - 6x + 9$ can be rewritten as a squared binomial: $(x-3)^2$.

Our detailed work confirms that the representation of the function is $(x-3)^2$. This matches with choice 4.

Therefore, the product representation of the function is $\boldsymbol{(x-3)^2}$.

To solve this problem, let's follow these steps:

• Step 1: Identify the coefficients: $a = 1$, $b = -3$, $c = -4$.
• Step 2: Calculate the product $ac = 1 \times -4 = -4$ and the sum $b = -3$.
• Step 3: Find two numbers that multiply to $-4$ and add to $-3$. These numbers are $-4$ and $1$.
• Step 4: Rewrite the middle term $-3x$ using $-4$ and $1$ as $-4x + x$.
• Step 5: Factor by grouping: $(x^2 - 4x) + (x - 4)$.
• Step 6: Factor out common factors: $x(x - 4) + 1(x - 4)$.
• Step 7: Notice the common binomial factor: $(x - 4)(x + 1)$.

Therefore, the factored form of the quadratic function $f(x) = x^2 - 3x - 4$ is $(x - 4)(x + 1)$, which corresponds to choice 3.

To solve this problem, we need to find a product representation of the quadratic function $f(x) = x^2 + 11x + 28$.

Let's go through the problem-solving process step-by-step:

Now let's address the problem:

To find the correct binomial factors of $f(x) = x^2 + 11x + 28$, we are searching for two numbers that multiply to 28 and add to 11. Examining possible pairs, $4$ and $7$ meet these criteria: $4 \times 7 = 28$ and $4 + 7 = 11$.

Thus, the quadratic expression can be rewritten as:

$f(x) = (x + 4)(x + 7)$

This checks our desired conditions. Multiplication confirms: $(x+4)(x+7) = x^2 + 7x + 4x + 28 = x^2 + 11x + 28$, which matches the original function.

Therefore, the product representation of $f(x) = x^2 + 11x + 28$ is $(x+7)(x+4)$.