Advanced Polynomial Operations

Multiplying and dividing polynomials

Advanced Polynomial Operations

Multiplying Polynomials

Use the distributive property repeatedly.

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

Distribute $2x$ : $2x(x^2 - 4x + 5) = 2x^3 - 8x^2 + 10x$

Distribute $3$ : $3(x^2 - 4x + 5) = 3x^2 - 12x + 15$

Combine: $2x^3 - 8x^2 + 10x + 3x^2 - 12x + 15$ $= 2x^3 - 5x^2 - 2x + 15$

Long Division of Polynomials

Similar to long division with numbers.

Steps:

Divide the leading terms
Multiply and subtract
Bring down the next term
Repeat until done

Synthetic Division

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

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

Remainder Theorem

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

📚 Practice Problems

1Problem 1easy

❓ Question:

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

💡 Show Solution

Distribute each term in the first polynomial:

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

$= x^3 + 3x^2 + 2x + 5x^2 + 15x + 10$

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

Answer: $x^3 + 8x^2 + 17x + 10$

2Problem 2medium

❓ Question:

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

💡 Show Solution

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

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

Answer: Remainder = $2$

3Problem 3hard

❓ Question:

Divide using long division: $(2x^3 + 3x^2 - 5x + 1) \div (x + 2)$

💡 Show Solution

Set up long division:

Step 1: $2x^3 \div x = 2x^2$ Multiply: $2x^2(x + 2) = 2x^3 + 4x^2$ Subtract: $(2x^3 + 3x^2) - (2x^3 + 4x^2) = -x^2$

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

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

Answer: $2x^2 - x - 3 + \frac{7}{x + 2}$

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