Let's solve each problem step by step from the worksheet titled
"Who Found It? Adding, Subtracting & Multiplying Polynomials".
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1. $(9x + x^2 - 6) + (-10x + 7 - 2x^2)$
Step 1: Remove parentheses:
$$
9x + x^2 - 6 - 10x + 7 - 2x^2
$$
Step 2: Combine like terms:
- $x^2 - 2x^2 = -x^2$
- $9x - 10x = -x$
- $-6 + 7 = 1$
Answer: $\boxed{-x^2 - x + 1}$
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2. $(9x^3 + 5x^2 + 11x) + (-2x^3 + 9x - 8x^2)$
Step 1: Remove parentheses:
$$
9x^3 + 5x^2 + 11x - 2x^3 + 9x - 8x^2
$$
Step 2: Combine like terms:
- $9x^3 - 2x^3 = 7x^3$
- $5x^2 - 8x^2 = -3x^2$
- $11x + 9x = 20x$
Answer: $\boxed{7x^3 - 3x^2 + 20x}$
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3. $(4x^3 - 6x^2 + x) + (2x^2 - 3x^2 - 7)$
Wait — there's a typo in the second polynomial: $2x^2 - 3x^2$ is already simplified. Let's rewrite it properly.
Actually, the expression is:
$$
(4x^3 - 6x^2 + x) + (2x^2 - 3x^2 - 7)
$$
Simplify inside the second parentheses:
$$
2x^2 - 3x^2 = -x^2
\Rightarrow (4x^3 - 6x^2 + x) + (-x^2 - 7)
$$
Now combine:
- $4x^3$
- $-6x^2 - x^2 = -7x^2$
- $x$
- $-7$
Answer: $\boxed{4x^3 - 7x^2 + x - 7}$
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4. $(12x^2 - 2x + 1) - (-13x^2 + 2x + 50)$
Step 1: Distribute the negative sign:
$$
12x^2 - 2x + 1 + 13x^2 - 2x - 50
$$
Step 2: Combine like terms:
- $12x^2 + 13x^2 = 25x^2$
- $-2x - 2x = -4x$
- $1 - 50 = -49$
Answer: $\boxed{25x^2 - 4x - 49}$
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5. $(10x^2 - 4x - 18x^3) - (10x - 2x^2 - 14x^3)$
Step 1: Rewrite first polynomial in standard form:
$$
-18x^3 + 10x^2 - 4x
$$
Distribute the negative sign:
$$
-18x^3 + 10x^2 - 4x - 10x + 2x^2 + 14x^3
$$
Step 2: Combine like terms:
- $-18x^3 + 14x^3 = -4x^3$
- $10x^2 + 2x^2 = 12x^2$
- $-4x - 10x = -14x$
Answer: $\boxed{-4x^3 + 12x^2 - 14x}$
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6. $(8x^2 + 6 - 3x^2) - (x^4 + 7x^2 - 3)$
Step 1: Simplify first parentheses:
$$
8x^2 - 3x^2 + 6 = 5x^2 + 6
$$
Now subtract:
$$
(5x^2 + 6) - (x^4 + 7x^2 - 3)
= 5x^2 + 6 - x^4 - 7x^2 + 3
$$
Step 2: Combine like terms:
- $-x^4$
- $5x^2 - 7x^2 = -2x^2$
- $6 + 3 = 9$
Answer: $\boxed{-x^4 - 2x^2 + 9}$
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7. $(2x^2 - 1)(3x + 7)$
Use distributive property (FOIL):
$$
2x^2 \cdot 3x = 6x^3 \\
2x^2 \cdot 7 = 14x^2 \\
-1 \cdot 3x = -3x \\
-1 \cdot 7 = -7
$$
Add them:
$$
6x^3 + 14x^2 - 3x - 7
$$
Answer: $\boxed{6x^3 + 14x^2 - 3x - 7}$
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8. $(5x - 7)(5x + 7)$
This is a
difference of squares pattern:
$$
(a - b)(a + b) = a^2 - b^2
$$
Here, $a = 5x$, $b = 7$
So:
$$
(5x)^2 - (7)^2 = 25x^2 - 49
$$
Answer: $\boxed{25x^2 - 49}$
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✔ Final Answers:
1. $-x^2 - x + 1$
2. $7x^3 - 3x^2 + 20x$
3. $4x^3 - 7x^2 + x - 7$
4. $25x^2 - 4x - 49$
5. $-4x^3 + 12x^2 - 14x$
6. $-x^4 - 2x^2 + 9$
7. $6x^3 + 14x^2 - 3x - 7$
8. $25x^2 - 49$
Let me know if you'd like these written neatly or formatted for printing!
Parent Tip: Review the logic above to help your child master the concept of adding subtracting multiplying and dividing polynomials worksheet.
