Factoring by Short Multiplication Practice Problems

Master factoring using abbreviated multiplication formulas with step-by-step practice problems. Learn difference of squares and perfect square trinomials.

📚Practice Factoring with Short Multiplication Formulas
  • Apply the difference of squares formula: a²-b²=(a-b)(a+b)
  • Factor perfect square trinomials using a²+2ab+b²=(a+b)²
  • Identify when to use a²-2ab+b²=(a-b)² formula
  • Solve factoring problems step-by-step with guided solutions
  • Master the three-step verification process for trinomial factoring
  • Build confidence with expressions like x²-64 and x²-18x+81

Understanding Factoring by Short Multiplication

Complete explanation with examples

The formulas for shortened multiplication will help us convert expressions with terms that have among them signs of addition or subtraction into expressions whose terms are multiplied.
The formulas for contracted multiplication are:
a2b2=(ab)×(a+b)a^2-b^2=(a-b)\times (a+b)

a2+2ab+b2=(a+b)2a^2+2ab+b^2=(a+b)^2

a22ab+b2=(ab)2a^2-2ab+b^2=(a-b)^2

Detailed explanation

Practice Factoring by Short Multiplication

Test your knowledge with 1 quizzes

Find the value of the parameter x.

\( x^2+x=0 \)

Examples with solutions for Factoring by Short Multiplication

Step-by-step solutions included
Exercise #1

Find the value of the parameter x.

9x+3x2=0 -9x+3x^2=0

Step-by-Step Solution

To solve the equation 9x+3x2=0 -9x + 3x^2 = 0 , follow these steps:

Step 1: Factor the equation.

Observe that both terms in the equation share a common factor, 3x 3x . We can factor this out:

9x+3x2=3x(3+x)-9x + 3x^2 = 3x(-3 + x).

The factored equation is 3x(3+x)=0 3x(-3 + x) = 0 .

Step 2: Apply the zero product property.

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two equations to solve:

  • 3x=0 3x = 0
  • 3+x=0 -3 + x = 0

Step 3: Solve each equation.

  • For 3x=0 3x = 0 , divide both sides by 3 to solve for x x :
  • x=0 x = 0
  • For 3+x=0 -3 + x = 0 , add 3 to both sides to solve for x x :
  • x=3 x = 3

Therefore, the solutions to the equation 9x+3x2=0 -9x + 3x^2 = 0 are x=0 x = 0 and x=3 x = 3 .

Matching these solutions to the given choices, the correct answer is choice 3: x=0,x=3 x = 0, x = 3 .

Thus, the values of x x that satisfy the equation are x=0 x = 0 and x=3 x = 3 .

Answer:

x=0,x=3 x=0,x=3

Video Solution
Exercise #2

Find the value of the parameter x.

x2+x=0 x^2+x=0

Step-by-Step Solution

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

  • Step 1: Factor the equation x2+x=0 x^2 + x = 0 .
  • Step 2: Use the Zero-Product Property to solve for x x .

Now, let's work through each step:

Step 1: Start by factoring the left-hand side of the equation:
x2+x=x(x+1) x^2 + x = x(x + 1)

Step 2: Apply the Zero-Product Property:
Since x(x+1)=0 x(x + 1) = 0 , we have two possible equations:
1) x=0 x = 0
2) x+1=0 x + 1 = 0

For the second equation, solve for x x :
x+1=0 x + 1 = 0 implies x=1 x = -1

Therefore, the solutions to the equation are x=0 x = 0 and x=1 x = -1 .

Hence, the value of the parameter x x is x=0,x=1 x = 0, x = -1 .

Answer:

x=0,x=1 x=0,x=-1

Video Solution
Exercise #3

Find the value of the parameter x.

9x312x2=0 9x^3-12x^2=0

Step-by-Step Solution

To solve this problem, we will factor the given polynomial expression:

Step 1: Identify the greatest common factor (GCF) in the equation 9x312x2=0 9x^3 - 12x^2 = 0 . The GCF of the terms 9x3 9x^3 and 12x2 12x^2 is 3x2 3x^2 .

Step 2: Factor out the GCF from the polynomial:

9x312x2=3x2(3x4)=0 9x^3 - 12x^2 = 3x^2(3x - 4) = 0 .

Step 3: Apply the zero-product property. Set each factor equal to zero:

  • 3x2=0 3x^2 = 0
  • 3x4=0 3x - 4 = 0

Step 4: Solve each equation for x x :

For 3x2=0 3x^2 = 0 , divide by 3:

x2=0 x^2 = 0 x=0 x = 0 .

For 3x4=0 3x - 4 = 0 , add 4 to both sides and then divide by 3:

3x=4 3x = 4
x=43 x = \frac{4}{3} .

Thus, the solutions to the equation 9x312x2=0 9x^3 - 12x^2 = 0 are x=0 x = 0 and x=43 x = \frac{4}{3} .

Therefore, the correct answer is:

x=0,x=43 x=0, x=\frac{4}{3}

Answer:

x=0,x=43 x=0,x=\frac{4}{3}

Video Solution
Exercise #4

Find the value of the parameter x.

x26x+8=0 x^2-6x+8=0

Step-by-Step Solution

To solve this quadratic equation by factoring, follow these steps:

  • Step 1: Write the equation: x26x+8=0 x^2 - 6x + 8 = 0 .
  • Step 2: Find two numbers that multiply to +8 +8 (the constant term) and add up to 6 -6 (the coefficient of x x ).

These numbers are 2 -2 and 4 -4 , since (2)×(4)=8 (-2) \times (-4) = 8 and (2)+(4)=6 (-2) + (-4) = -6 .

  • Step 3: Rewrite the quadratic expression as (x2)(x4)=0 (x - 2)(x - 4) = 0 .
  • Step 4: Solve each factor separately:
    • x2=0x=2 x - 2 = 0 \Rightarrow x = 2
    • x4=0x=4 x - 4 = 0 \Rightarrow x = 4

Therefore, the solutions to the quadratic equation x26x+8=0 x^2 - 6x + 8 = 0 are x=2 x = 2 and x=4 x = 4 .

The correct choice for the solution is:

x=2,x=4 x=2,x=4 which corresponds to choice 4.

Answer:

x=2,x=4 x=2,x=4

Video Solution

Frequently Asked Questions

What are the three short multiplication formulas for factoring?

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The three abbreviated multiplication formulas are: 1) a²-b²=(a-b)(a+b) for difference of squares, 2) a²+2ab+b²=(a+b)² for perfect square trinomials with addition, and 3) a²-2ab+b²=(a-b)² for perfect square trinomials with subtraction.

How do I know when to use the difference of squares formula?

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Use a²-b²=(a-b)(a+b) when you have two conditions: 1) One positive and one negative term, and 2) You can find the square root of each term separately. For example, x²-64 becomes (x-8)(x+8).

What are the steps to factor a perfect square trinomial?

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Follow these three steps: 1) Check if the first and last terms have the same sign (both positive or both negative), 2) Verify you can take the square root of both terms, 3) Confirm that 2×(root1)×(root2) equals the middle term. If all conditions are met, use the appropriate formula.

Why can't I factor x²+2x-1 using short multiplication formulas?

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You cannot use abbreviated multiplication formulas when the first and last terms have different signs (one positive, one negative) for trinomials. The difference of squares formula only works with two terms, not three.

What's the difference between (a+b)² and (a-b)² formulas?

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Both formulas factor perfect square trinomials, but (a+b)²=a²+2ab+b² has a positive middle term, while (a-b)²=a²-2ab+b² has a negative middle term. The sign of the middle term in your original expression determines which formula to use.

How do I factor x²-18x+81 step by step?

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Step 1: Both x² and 81 are positive ✓. Step 2: √x²=x and √81=9 ✓. Step 3: 2×x×9=18x matches the middle term (with negative sign) ✓. Since all conditions are met and the middle term is negative, use (a-b)²: (x-9)².

What are common mistakes when factoring by short multiplication?

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Common errors include: 1) Using the wrong formula when signs don't match the pattern, 2) Forgetting to check if terms are perfect squares, 3) Not verifying the middle term equals 2ab, and 4) Mixing up positive and negative signs in the final factored form.

Can I use short multiplication formulas for expressions with more than three terms?

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Short multiplication formulas work best with two terms (difference of squares) or three terms (perfect square trinomials). For expressions with more terms, you may need to group terms first or use other factoring methods like factoring by grouping or finding common factors.

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