Advanced Polynomial Operations

Multiplying and dividing polynomials

🎯⭐ INTERACTIVE LESSON

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Advanced Polynomial Operations

Multiplying Polynomials

Use the distributive property repeatedly.

Example: (2x+3)(x2−4x+5)(2x + 3)(x^2 - 4x + 5)

Distribute 2x2x: 2x(x2−4x+5)=2x3−8x2+10x2x(x^2 - 4x + 5) = 2x^3 - 8x^2 + 10x

Distribute 33: 3(x2−4x+5)=3x2−12x+153(x^2 - 4x + 5) = 3x^2 - 12x + 15

Combine: 2x3−8x2+10x+3x2−12x+152x^3 - 8x^2 + 10x + 3x^2 - 12x + 15 =2x3−5x2−2x+15= 2x^3 - 5x^2 - 2x + 15

Long Division of Polynomials

Similar to long division with numbers.

Steps:

  1. Divide the leading terms
  2. Multiply and subtract
  3. Bring down the next term
  4. Repeat until done

Synthetic Division

A shortcut for dividing by (x−c)(x - c).

Use only when divisor is in form (x−c)(x - c).

Remainder Theorem

When polynomial P(x)P(x) is divided by (x−c)(x - c): Remainder=P(c)\text{Remainder} = P(c)

📚 Practice Problems

1Problem 1easy

❓ Question:

Multiply: (x+5)(x2+3x+2)(x + 5)(x^2 + 3x + 2)

💡 Show Solution

Distribute each term in the first polynomial:

x(x2+3x+2)+5(x2+3x+2)x(x^2 + 3x + 2) + 5(x^2 + 3x + 2)

=x3+3x2+2x+5x2+15x+10= x^3 + 3x^2 + 2x + 5x^2 + 15x + 10

Combine like terms: =x3+8x2+17x+10= x^3 + 8x^2 + 17x + 10

Answer: x3+8x2+17x+10x^3 + 8x^2 + 17x + 10

2Problem 2medium

❓ Question:

Use the Remainder Theorem to find the remainder when P(x)=x3−4x2+6x−2P(x) = x^3 - 4x^2 + 6x - 2 is divided by (x−2)(x - 2)

💡 Show Solution

By the Remainder Theorem, the remainder when dividing by (x−2)(x - 2) is P(2)P(2).

Evaluate P(2)P(2): P(2)=(2)3−4(2)2+6(2)−2P(2) = (2)^3 - 4(2)^2 + 6(2) - 2 =8−16+12−2= 8 - 16 + 12 - 2 =2= 2

Answer: Remainder = 22

3Problem 3hard

❓ Question:

Divide using long division: (2x3+3x2−5x+1)÷(x+2)(2x^3 + 3x^2 - 5x + 1) \div (x + 2)

💡 Show Solution

Set up long division:

Step 1: 2x3÷x=2x22x^3 \div x = 2x^2 Multiply: 2x2(x+2)=2x3+4x22x^2(x + 2) = 2x^3 + 4x^2 Subtract: (2x3+3x2)−(2x3+4x2)=−x2(2x^3 + 3x^2) - (2x^3 + 4x^2) = -x^2

Step 2: −x2÷x=−x-x^2 \div x = -x Multiply: −x(x+2)=−x2−2x-x(x + 2) = -x^2 - 2x Subtract: (−x2−5x)−(−x2−2x)=−3x(-x^2 - 5x) - (-x^2 - 2x) = -3x

Step 3: −3x÷x=−3-3x \div x = -3 Multiply: −3(x+2)=−3x−6-3(x + 2) = -3x - 6 Subtract: (−3x+1)−(−3x−6)=7(-3x + 1) - (-3x - 6) = 7

Answer: 2x2−x−3+7x+22x^2 - x - 3 + \frac{7}{x + 2}

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Factoring Polynomials
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