Let’s solve each problem step by step. We’ll use the distributive property (also called FOIL for binomials) to multiply polynomials.
---
Problem 1: (a + 4)(a + 7)
Multiply each term in the first parentheses by each term in the second:
= a·a + a·7 + 4·a + 4·7
= a² + 7a + 4a + 28
Combine like terms:
=
a² + 11a + 28
---
Problem 2: (a - 7)(a - 3)
= a·a + a·(-3) + (-7)·a + (-7)·(-3)
= a² - 3a - 7a + 21
Combine like terms:
=
a² - 10a + 21
---
Problem 3: (a - 7)(a - 5)
= a·a + a·(-5) + (-7)·a + (-7)·(-5)
= a² - 5a - 7a + 35
Combine like terms:
=
a² - 12a + 35
---
Problem 4: (3x - 4)(2x - 8)
= 3x·2x + 3x·(-8) + (-4)·2x + (-4)·(-8)
= 6x² - 24x - 8x + 32
Combine like terms:
=
6x² - 32x + 32
---
Problem 5: (x - 2)(x + 1)
= x·x + x·1 + (-2)·x + (-2)·1
= x² + x - 2x - 2
Combine like terms:
=
x² - x - 2
---
Problem 6: (x - 1)(x² + 6x - 1)
Distribute x and then -1 across the trinomial:
= x·(x² + 6x - 1) + (-1)·(x² + 6x - 1)
= x³ + 6x² - x - x² - 6x + 1
Now combine like terms:
x³ + (6x² - x²) + (-x - 6x) + 1
=
x³ + 5x² - 7x + 1
---
Problem 7: (x - 8)(x - 2)
= x·x + x·(-2) + (-8)·x + (-8)·(-2)
= x² - 2x - 8x + 16
Combine like terms:
=
x² - 10x + 16
---
Problem 8: (x + 6)(x - 3)
= x·x + x·(-3) + 6·x + 6·(-3)
= x² - 3x + 6x - 18
Combine like terms:
=
x² + 3x - 18
---
Problem 9: (4x + 1)(3x - 4)
= 4x·3x + 4x·(-4) + 1·3x + 1·(-4)
= 12x² - 16x + 3x - 4
Combine like terms:
=
12x² - 13x - 4
---
Problem 10: (2x - 6)(x + 2)
= 2x·x + 2x·2 + (-6)·x + (-6)·2
= 2x² + 4x - 6x - 12
Combine like terms:
=
2x² - 2x - 12
---
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 polynomials multiplication worksheet.
