Find Missing Terms in (?-?)(8y-7) = 8xy-32y-7x+28: Polynomial Expansion

Q: Why does the order of terms not matter in the final answer?

Because addition is commutative, meaning $ a + b = b + a $. So $ 8xy - 7x - 32y + 28 $ equals $ 8xy - 32y - 7x + 28 $ - they're the same expression!

Q: How do I know which terms go in the first factor?

Look for common factors in the expanded form. Since we see $ 8xy $ and $ -7x $, the first factor likely contains x. The coefficients help determine the second term.

Q: What if I get (4-x) instead of (x-4)?

That's actually equivalent! Remember that $ (4-x) = -(x-4) $. If you expand both, you'll get the same result, just with opposite signs that cancel out properly.

Q: How can I check my factorization is correct?

Always expand your answer! Take your factored form like $ (x-4)(8y-7) $ and multiply it out using FOIL or distributive property. If it matches the given expression, you're right!

Q: Why is the answer written as (-4, x) instead of (x, -4)?

The format $ (?-?) $ suggests the first blank comes first. Since we have $ x-4 $, the blanks would be filled as x then -4, giving us the ordered pair format.

Polynomial Factorization with Missing Terms

Fill in the missing values:

$(?-?)(8y-7)=8xy-32y-7x+28$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Complete the missing values

00:10 Let's factor 32 into factors 8 and 4

00:16 Let's factor 28 into factors 7 and 4

00:23 Let's mark the common factors

00:50 Let's take out the common factors from the parentheses

01:14 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Fill in the missing values:

$(?-?)(8y-7)=8xy-32y-7x+28$

Step-by-step solution

To solve this problem, we will expand the left-hand expression and set it equal to the right-hand side.

Let's rewrite the expression: $(x-4)(8y-7)$ .

We begin by expanding the left-hand side using the distributive property:
$(x-4)(8y-7) = x(8y-7) - 4(8y-7)$ .
Further expand each component:
$x(8y-7) = 8xy - 7x$ and $-4(8y-7) = -32y + 28$ .
Combine to form:
$8xy - 7x - 32y + 28$ .

We compare this with the right-hand side of the original equation, $8xy - 32y - 7x + 28$ .

The expressions match perfectly upon comparative structuring.

Hence, the missing expression in $(x-4)$ confirms accurate factorization.

Thus, the missing values that satisfy the equation are $x-4$ .

Therefore, the missing values are $(-4, x)$ or equivalently $x, -4$ .

Final Answer

$(-4,x)$ $x,-4$

Key Points to Remember

Essential concepts to master this topic

Factorization Rule: Use distributive property to expand and compare coefficients
Technique: Match $8xy - 7x - 32y + 28$ with given expression
Check: Verify $(x-4)(8y-7) = 8xy - 32y - 7x + 28$ ✓

Common Mistakes

Avoid these frequent errors

Assuming terms must be in same order
Don't expect $8xy - 7x - 32y + 28$ to match exactly with $8xy - 32y - 7x + 28$ = confusion about equality! Terms can be rearranged since addition is commutative. Always focus on matching all terms regardless of their order.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why does the order of terms not matter in the final answer?

Because addition is commutative, meaning $a + b = b + a$ . So $8xy - 7x - 32y + 28$ equals $8xy - 32y - 7x + 28$ - they're the same expression!

How do I know which terms go in the first factor?

Look for common factors in the expanded form. Since we see $8xy$ and $-7x$ , the first factor likely contains x. The coefficients help determine the second term.

What if I get (4-x) instead of (x-4)?

That's actually equivalent! Remember that $(4-x) = -(x-4)$ . If you expand both, you'll get the same result, just with opposite signs that cancel out properly.

How can I check my factorization is correct?

Always expand your answer! Take your factored form like $(x-4)(8y-7)$ and multiply it out using FOIL or distributive property. If it matches the given expression, you're right!

Why is the answer written as (-4, x) instead of (x, -4)?

The format $(?-?)$ suggests the first blank comes first. Since we have $x-4$ , the blanks would be filled as x then -4, giving us the ordered pair format.

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