Question

date: ____ bell: ____ unit 7: polynomials & factoring
homework 2: monomial × polynomial
directions: simplify the following polynomials. answers must be in standard form.

1. ( a(4a + 3) )
2. ( x(11x + 4) )
3. ( x(2x - 5) )
4. ( 2y^3(y^5 - 9) )
5. ( -3mn(m^2n^3 + 2mn) )
6. ( 9p^3q^2(3p^3 - 5q) )
7. ( 3r^2(5r^2 - r + 4) )
8. ( 7c(c^3 - 2c^2 + 5) )
9. ( -3n^2(-2n^3 + 7n + 4n) )
10. ( w^3(3w^2 + 2w) + 5w^4 )
11. ( x(5x - 3) - 2x )
12. ( y^2(-4y + 5) - 6y^2 )
13. ( 2x(3x^2 + 4) - 3x^2 )
14. ( 4a(5a^2 - 4) + 9a )
15. ( -2(4v^3 + 5v) + v(v^2 + 6v) )
16. ( 4b(5b - 3) - 2(b^2 - 7b - 4) )
17. ( 4n(3n^2 + n - 4) - n(3 - n) )
18. ( 3m(3m + 6) - 3(m^2 + 4m + 1) )
19. ( 2(4k^2 - 2k) - 3(-6k^2 + 4) + 2k(k - 1) )
20. write an expression in simplest form to represent the area of the shaded region.

image of a shaded region with outer rectangle dimensions ( 4x ) (length) and ( 3x + 2 ) (width), and inner rectangle dimensions ( 3x ) (length) and ( 2x ) (width)

Explanation:

Step1: Distribute monomial to terms

$a(4a + 3) = 4a^2 + 3a$

Step2: Distribute negative monomial

$-c(11c + 4) = -11c^2 - 4c$

Step3: Distribute monomial to terms

$x(2x - 5) = 2x^2 - 5x$

Step4: Distribute monomial to terms

$2y^2(y^5 - 4y) = 2y^7 - 8y^3$

Step5: Distribute monomial to terms

$-3mn(m^2n^3 + 2mn) = -3m^3n^4 - 6m^2n^2$

Step6: Distribute monomial to terms

$9p^3q^4(3p^3 - 5q) = 27p^6q^4 - 45p^3q^5$

Step7: Distribute monomial to terms

$3r^4(5r^2 - r + 4) = 15r^6 - 3r^5 + 12r^4$

Step8: Distribute monomial to terms

$7c(c^3 - 2c^2 + 5) = 7c^4 - 14c^3 + 35c$

Step9: Distribute monomial to terms

$-3n^2(-2n^3 + 7n + 4n) = 6n^5 - 33n^3$

Step10: Distribute then combine like terms

$w^3(3w^2 + 2w) + 5w^4 = 3w^5 + 2w^4 + 5w^4 = 3w^5 + 7w^4$

Step11: Distribute then combine like terms

$z(5z - 3) - 2z = 5z^2 - 3z - 2z = 5z^2 - 5z$

Step12: Distribute then combine like terms

$y^2(-4y + 5) - 6y^2 = -4y^3 + 5y^2 - 6y^2 = -4y^3 - y^2$

Step13: Distribute then combine like terms

$2x(3x^2 + 4) - 3x^2 = 6x^3 + 8x - 3x^2 = 6x^3 - 3x^2 + 8x$

Step14: Distribute then combine like terms

$4a(5a^2 - 4) + 9a = 20a^3 - 16a + 9a = 20a^3 - 7a$

Step15: Distribute then combine like terms

$-2(4v^2 + 5v) + v(v^2 + 6v) = -8v^2 - 10v + v^3 + 6v^2 = v^3 - 2v^2 - 10v$

Step16: Distribute then combine like terms

$4b(5b - 3) - 2(b^2 - 7b - 4) = 20b^2 - 12b - 2b^2 + 14b + 8 = 18b^2 + 2b + 8$

Step17: Distribute then combine like terms

$4n(3n^2 + n - 4) - n(3 - n) = 12n^3 + 4n^2 - 16n - 3n + n^2 = 12n^3 + 5n^2 - 19n$

Step18: Distribute then combine like terms

$3m(3m + 6) - 3(m^2 + 4m + 1) = 9m^2 + 18m - 3m^2 - 12m - 3 = 6m^2 + 6m - 3$

Step19: Distribute all terms then combine like terms

$2(4k^2 - 2k) - 3(-6k^2 + 4) + 2k(k - 1) = 8k^2 - 4k + 18k^2 - 12 + 2k^2 - 2k = 28k^2 - 6k - 12$

Step20: Find area of large rectangle, subtract inner rectangle

$\text{Area of large rectangle} = 4x(3x+2) = 12x^2 + 8x$
$\text{Area of inner rectangle} = 3x(2x) = 6x^2$
$\text{Shaded Area} = (12x^2 + 8x) - 6x^2 = 6x^2 + 8x$

Answer:

1. $4a^2 + 3a$
2. $-11c^2 - 4c$
3. $2x^2 - 5x$
4. $2y^7 - 8y^3$
5. $-3m^3n^4 - 6m^2n^2$
6. $27p^6q^4 - 45p^3q^5$
7. $15r^6 - 3r^5 + 12r^4$
8. $7c^4 - 14c^3 + 35c$
9. $6n^5 - 33n^3$
10. $3w^5 + 7w^4$
11. $5z^2 - 5z$
12. $-4y^3 - y^2$
13. $6x^3 - 3x^2 + 8x$
14. $20a^3 - 7a$
15. $v^3 - 2v^2 - 10v$
16. $18b^2 + 2b + 8$
17. $12n^3 + 5n^2 - 19n$
18. $6m^2 + 6m - 3$
19. $28k^2 - 6k - 12$
20. $6x^2 + 8x$