To solve these polynomial multiplication problems, we use the distributive property — often called FOIL for binomials (First, Outer, Inner, Last). We multiply each term in the first parentheses by each term in the second, then combine like terms.
Let’s go one by one:
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1. (a + 4)(a + 7)
= a·a + a·7 + 4·a + 4·7
= a² + 7a + 4a + 28
= a² + 11a + 28
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2. (a - 7)(a - 3)
= a·a + a·(-3) + (-7)·a + (-7)·(-3)
= a² - 3a - 7a + 21
= a² - 10a + 21
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3. (a - 7)(a - 5)
= a·a + a·(-5) + (-7)·a + (-7)·(-5)
= a² - 5a - 7a + 35
= a² - 12a + 35
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4. (3x - 4)(2x - 8)
= 3x·2x + 3x·(-8) + (-4)·2x + (-4)·(-8)
= 6x² - 24x - 8x + 32
= 6x² - 32x + 32
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5. (x - 2)(x + 1)
= x·x + x·1 + (-2)·x + (-2)·1
= x² + x - 2x - 2
= x² - x - 2
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6. (x - 1)(x² + 6x - 1)
This is a binomial times a trinomial. Distribute each term:
x·(x² + 6x - 1) = x³ + 6x² - x
-1·(x² + 6x - 1) = -x² - 6x + 1
Now add them:
x³ + 6x² - x - x² - 6x + 1
= x³ + (6x² - x²) + (-x - 6x) + 1
= x³ + 5x² - 7x + 1
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7. (x - 8)(x - 2)
= x·x + x·(-2) + (-8)·x + (-8)·(-2)
= x² - 2x - 8x + 16
= x² - 10x + 16
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8. (x + 6)(x - 3)
= x·x + x·(-3) + 6·x + 6·(-3)
= x² - 3x + 6x - 18
= x² + 3x - 18
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9. (4x + 1)(3x - 4)
= 4x·3x + 4x·(-4) + 1·3x + 1·(-4)
= 12x² - 16x + 3x - 4
= 12x² - 13x - 4
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10. (2x - 6)(x + 2)
= 2x·x + 2x·2 + (-6)·x + (-6)·2
= 2x² + 4x - 6x - 12
= 2x² - 2x - 12
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All calculations checked. Final answers are ready.
Final Answer:
1. a² + 11a + 28
2. a² - 10a + 21
3. a² - 12a + 35
4. 6x² - 32x + 32
5. x² - x - 2
6. x³ + 5x² - 7x + 1
7. x² - 10x + 16
8. x² + 3x - 18
9. 12x² - 13x - 4
10. 2x² - 2x - 12
Parent Tip: Review the logic above to help your child master the concept of multiplying polynomials practice worksheet.
