Multiplying Polynomials: Practice

Multiply each term by each term, tracking the negatives.

$6x^2 - 15x - 4x + 10 = 6x^2 - 19x + 10$

The last product $(-2)(-5) = +10$ is positive.

Distribute $x$ and then $4$ across the trinomial, giving six products.

$x^3 - 3x^2 + 2x + 4x^2 - 12x + 8$

Combine like terms: $-3x^2 + 4x^2 = x^2$ and $2x - 12x = -10x$, so the product is $x^3 + x^2 - 10x + 8$.

The cross terms cancel, leaving a difference of squares.

$4x^2 + 2x - 2x - 1 = 4x^2 - 1$

Square the binomial as $(x - 5)(x - 5)$; the middle term is twice the product of the parts.

$x^2 - 5x - 5x + 25 = x^2 - 10x + 25$

The degree of a product is the sum of the degrees; the leading terms multiply.

$x^2 \cdot x^3 = x^5$

Degree $2 + 3 = 5$.

The $x$ term comes from the outer and inner products, $4x$ and $3x$.

$4x + 3x = 7x$

So the coefficient of $x$ is $7$.

The constant term is the product of the two constants.

$(-3)(5) = -15$

Multiply $-2x$ by each term; watch the sign on the middle product.

$(-2x)(3x^2) + (-2x)(-x) + (-2x)(4) = -6x^3 + 2x^2 - 8x$

The product $(-2x)(-x) = +2x^2$ is positive.

Square as $(2x + 5)(2x + 5)$; the middle term is twice the product of the parts.

$4x^2 + 10x + 10x + 25 = 4x^2 + 20x + 25$

The leading term of a product is the product of the leading terms.

$(2x^3)(4x^2) = 8x^5$

Distribute and watch the middle terms cancel in pairs.

$x^3 + 4x^2 + 16x - 4x^2 - 16x - 64 = x^3 - 64$

This is the difference-of-cubes pattern.

The cross terms $-7x$ and $+7x$ cancel.

$x^2 - 7x + 7x - 49 = x^2 - 49$