Solve the following algebraic expressions: 1. 4d^2(-2d^3 + 2d^2 - 6d + 2) 2. (2y + 3)(y - 5) 3. (b - 8)(5b - 2) 4. (-p + 2)(3p^3 + 1) 5. (-3d + 10)(2d - 1) 6. (w - 3)(w^2 + 8w + 1)... Solve the following algebraic expressions: 1. 4d^2(-2d^3 + 2d^2 - 6d + 2) 2. (2y + 3)(y - 5) 3. (b - 8)(5b - 2) 4. (-p + 2)(3p^3 + 1) 5. (-3d + 10)(2d - 1) 6. (w - 3)(w^2 + 8w + 1) 7. (x^3 - 4xy + y^2)(5xy)

Understand the Problem

The user has provided a set of math problems, each involving the multiplication of polynomials. These problems require the student to apply the distributive property (often referred to as the FOIL method for binomials) and combine like terms to simplify the expressions. Each question should be solved step by step, showing the distribution and simplification process.

Answer

3. $-8d^5 + 8d^4 - 24d^3 + 8d^2$ 4. $2y^2 - 7y - 15$ 5. $5b^2 - 42b + 16$ 6. $-3p^4 + 6p^3 - p + 2$ 7. $-6d^2 + 23d - 10$ 8. $w^3 + 5w^2 - 23w - 3$ 9. $5x^4y - 20x^2y^2 + 5xy^3$

Steps to Solve

Problem 3: Distribute $4d^2$

Multiply $4d^2$ by each term inside the parentheses: $4d^2(-2d^3 + 2d^2 - 6d + 2) = 4d^2(-2d^3) + 4d^2(2d^2) + 4d^2(-6d) + 4d^2(2)$

Simplify the terms

Simplify each term by multiplying the coefficients and adding the exponents of $d$: $-8d^5 + 8d^4 - 24d^3 + 8d^2$

Problem 5: Expand $(2y+3)(y-5)$ using FOIL

Multiply the First, Outer, Inner, and Last terms: $(2y+3)(y-5) = (2y)(y) + (2y)(-5) + (3)(y) + (3)(-5)$

Simplify the terms

Simplify each term: $2y^2 - 10y + 3y - 15$

Combine like terms

Combine the $y$ terms: $2y^2 - 7y - 15$

Problem 7: Expand $(b-8)(5b-2)$ using FOIL

Multiply the First, Outer, Inner, and Last terms: $(b-8)(5b-2) = (b)(5b) + (b)(-2) + (-8)(5b) + (-8)(-2)$

Simplify the terms

Simplify each term: $5b^2 - 2b - 40b + 16$

Combine like terms

Combine the $b$ terms: $5b^2 - 42b + 16$

Problem 9: Expand $(-p+2)(3p^3+1)$ using FOIL

Multiply the First, Outer, Inner, and Last terms: $(-p+2)(3p^3+1) = (-p)(3p^3) + (-p)(1) + (2)(3p^3) + (2)(1)$

Simplify the terms

Simplify each term: $-3p^4 - p + 6p^3 + 2$

Rearrange in descending order of exponents

Rearrange the terms: $-3p^4 + 6p^3 - p + 2$

Problem 11: Expand $(-3d+10)(2d-1)$ using FOIL

Multiply the First, Outer, Inner, and Last terms: $(-3d+10)(2d-1) = (-3d)(2d) + (-3d)(-1) + (10)(2d) + (10)(-1)$

Simplify the terms

Simplify each term: $-6d^2 + 3d + 20d - 10$

Combine like terms

Combine the $d$ terms: $-6d^2 + 23d - 10$

Problem 13: Expand $(w-3)(w^2+8w+1)$ by distributing

Multiply each term in the first parenthesis by each term in the second: $(w-3)(w^2+8w+1) = w(w^2) + w(8w) + w(1) - 3(w^2) - 3(8w) - 3(1)$

Simplify the terms

Simplify each term: $w^3 + 8w^2 + w - 3w^2 - 24w - 3$

Combine like terms

Combine the $w^2$ and $w$ terms: $w^3 + 5w^2 - 23w - 3$

Problem 15: Distribute $5xy$

Multiply $5xy$ by each term inside the parentheses: $(x^3 - 4xy + y^2)(5xy) = 5xy(x^3) - 5xy(4xy) + 5xy(y^2)$

Simplify the terms

Simplify each term by multiplying the coefficients and adding the exponents: $5x^4y - 20x^2y^2 + 5xy^3$

$-8d^5 + 8d^4 - 24d^3 + 8d^2$
$2y^2 - 7y - 15$
$5b^2 - 42b + 16$
$-3p^4 + 6p^3 - p + 2$
$-6d^2 + 23d - 10$
$w^3 + 5w^2 - 23w - 3$
$5x^4y - 20x^2y^2 + 5xy^3$

More Information

Each problem involves polynomial multiplication. Remember to distribute carefully and combine like terms!

Tips

Forgetting to distribute to all terms inside the parentheses.
Incorrectly multiplying exponents. Remember that $x^a \cdot x^b = x^{a+b}$.
Incorrectly multiplying coefficients.
Forgetting to combine like terms.
Making sign errors (especially with negative numbers).

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