Multiplying Polynomials: Practice

Multiply the coefficients and add the exponents of the like base.

$(2 \cdot 3)\,x^{2+4} = 6x^6$

The exponents add to $6$; multiplying them to get $x^8$ is the classic slip.

Multiply the coefficients, then add the exponents ($x$ is $x^1$).

$(5 \cdot 4)\,x^{1+1} = 20x^2$

Fold the negative sign into the coefficient, then add the exponents.

$(-3 \cdot 2)\,x^{3+2} = -6x^5$

Multiply $x$ by each term. For the first product add the exponents: $x \cdot x = x^2$.

$x \cdot x + x \cdot 7 = x^2 + 7x$

Multiply $2x$ by each term, adding exponents on the first product.

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

Four products, then combine the like middle terms.

$x^2 + 5x + 4x + 20 = x^2 + 9x + 20$

Keep every sign in place through the four products.

$x^2 - 3x - 2x + 6 = x^2 - 5x + 6$

Two negatives multiply to the positive constant $+6$.

Multiply the coefficients and add the exponents.

$(6 \cdot 2)\,x^{3+1} = 12x^4$

The two middle products cancel: $-6x + 6x = 0$.

$x^2 - 6x + 6x - 36 = x^2 - 36$

This is the difference of squares pattern.

Squaring means $(x + 3)(x + 3)$, which has a middle term, not just $x^2 + 9$.

$x^2 + 3x + 3x + 9 = x^2 + 6x + 9$

Multiply the coefficients (one is negative) and add the exponents.

$(3 \cdot -2)\,x^{1+4} = -6x^5$

Multiply $4x$ by each of the three terms, adding exponents each time.

$(4x)(x^2) + (4x)(2x) + (4x)(3) = 4x^3 + 8x^2 + 12x$

The last product keeps its $x$: $4x \cdot 3 = 12x$, not $12$.