Question

simplify the following polynomials. all final answers must be in standard form. box answers.
$(3y^{2}-4y+1)+(y^{2}-y-2)$
$(2m^{2}-8m+2)-(3m^{2}+5m-7)$
$(7-4k^{2}+k)+(8k+5k^{2}-7)$
$(6x^{2}-5x)-(4x^{2}-3x+4)$
$3a(6a^{2}-4ab^{2})+8a^{2}b^{2}-2b^{3}$
$-2x(x^{3}-6x^{2}+6)+4x^{3}-(5x^{4}+10x)$
$(x-4)(x+6)$
$(2m-3)(6m+1)$
$(5a+b)(a-4b)$
$(7a-1)^{2}$

Explanation:

Step1: Combine like terms

$(3y^2 - 4y + 1) + (y^2 - y - 2) = 3y^2 + y^2 - 4y - y + 1 - 2$

Step2: Simplify the expression

$= 4y^2 - 5y - 1$

---

Step1: Distribute negative sign

$(2m^2 - 8m + 2) - (3m^2 + 5m - 7) = 2m^2 - 8m + 2 - 3m^2 - 5m + 7$

Step2: Combine like terms

$= 2m^2 - 3m^2 - 8m - 5m + 2 + 7$

Step3: Simplify the expression

$= -m^2 - 13m + 9$

---

Step1: Combine like terms

$(7 - 4k^2 + k) + (8k + 5k^2 - 7) = -4k^2 + 5k^2 + k + 8k + 7 - 7$

Step2: Simplify the expression

$= k^2 + 9k$

---

Step1: Distribute negative sign

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

Step2: Combine like terms

$= 6x^2 - 4x^2 - 5x + 3x - 4$

Step3: Simplify the expression

$= 2x^2 - 2x - 4$

---

Step1: Distribute the monomial

$3a(6a^2 - 4ab^2) + 8a^2b^2 - 2b^3 = 18a^3 - 12a^2b^2 + 8a^2b^2 - 2b^3$

Step2: Combine like terms

$= 18a^3 + (-12a^2b^2 + 8a^2b^2) - 2b^3$

Step3: Simplify the expression

$= 18a^3 - 4a^2b^2 - 2b^3$

---

Step1: Distribute the monomials

$-2x(x^3 - 6x^2 + 6) + 4x^3 - (5x^4 + 10x) = -2x^4 + 12x^3 - 12x + 4x^3 - 5x^4 - 10x$

Step2: Combine like terms

$= -2x^4 - 5x^4 + 12x^3 + 4x^3 - 12x - 10x$

Step3: Simplify the expression

$= -7x^4 + 16x^3 - 22x$

---

Step1: Use FOIL method

$(x - 4)(x + 6) = x^2 + 6x - 4x - 24$

Step2: Combine like terms

$= x^2 + 2x - 24$

---

Step1: Use FOIL method

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

Step2: Combine like terms

$= 12m^2 - 16m - 3$

---

Step1: Use FOIL method

$(5a + b)(a - 4b) = 5a^2 - 20ab + ab - 4b^2$

Step2: Combine like terms

$= 5a^2 - 19ab - 4b^2$

---

Step1: Use square of binomial rule

$(7a - 1)^2 = (7a)^2 - 2(7a)(1) + 1^2$

Step2: Simplify the expression

$= 49a^2 - 14a + 1$

Answer:

1. $4y^2 - 5y - 1$
2. $-m^2 - 13m + 9$
3. $k^2 + 9k$
4. $2x^2 - 2x - 4$
5. $18a^3 - 4a^2b^2 - 2b^3$
6. $-7x^4 + 16x^3 - 22x$
7. $x^2 + 2x - 24$
8. $12m^2 - 16m - 3$
9. $5a^2 - 19ab - 4b^2$
10. $49a^2 - 14a + 1$