Question

add/subtract the polynomials

1. $(4x^4 + 21x^2 - 15x - 6) + (18x + 7x^4 - 13x^2 + 3)$

1. $(7x^3 + 68x^4 - 14x + 1) - (-10x^3 + 8x + 23)$

multiply the polynomials

1. $(x - 3)(x^2 - 2x + 7)$

Explanation:

Problem 1: Add the Polynomials \((4x^4 + 21x^2 - 15x - 6) + (18x + 7x^4 - 13x^2 + 3)\)

Step 1: Combine like terms for \(x^4\)

\(4x^4 + 7x^4 = 11x^4\)

Step 2: Combine like terms for \(x^2\)

\(21x^2 - 13x^2 = 8x^2\)

Step 3: Combine like terms for \(x\)

\(-15x + 18x = 3x\)

Step 4: Combine constant terms

\(-6 + 3 = -3\)

Step 5: Combine all terms

\(11x^4 + 8x^2 + 3x - 3\)

Step 1: Distribute the negative sign

\(7x^3 + 68x^4 - 14x + 1 + 10x^3 - 8x - 23\)

Step 2: Combine like terms for \(x^4\)

\(68x^4\) (no other \(x^4\) terms)

Step 3: Combine like terms for \(x^3\)

\(7x^3 + 10x^3 = 17x^3\)

Step 4: Combine like terms for \(x\)

\(-14x - 8x = -22x\)

Step 5: Combine constant terms

\(1 - 23 = -22\)

Step 6: Combine all terms

\(68x^4 + 17x^3 - 22x - 22\)

Step 1: Distribute \(x\) to each term in the second polynomial

\(x \cdot x^2 = x^3\), \(x \cdot (-2x) = -2x^2\), \(x \cdot 7 = 7x\)

Step 2: Distribute \(-3\) to each term in the second polynomial

\(-3 \cdot x^2 = -3x^2\), \(-3 \cdot (-2x) = 6x\), \(-3 \cdot 7 = -21\)

Step 3: Combine all terms

\(x^3 - 2x^2 + 7x - 3x^2 + 6x - 21\)

Step 4: Combine like terms for \(x^2\)

\(-2x^2 - 3x^2 = -5x^2\)

Step 5: Combine like terms for \(x\)

\(7x + 6x = 13x\)

Step 6: Combine all terms

\(x^3 - 5x^2 + 13x - 21\)

Answer:

\(11x^4 + 8x^2 + 3x - 3\)

Problem 3: Subtract the Polynomials \((7x^3 + 68x^4 - 14x + 1) - (-10x^3 + 8x + 23)\)