Find the Product of Algebraic Expressions then Add or Subtract

This lesson combines two skills: multiplying algebraic expressions with the distributive law, and then adding or subtracting the products. Along the way it evaluates expressions at given values and uses both the column and row layouts to collect like terms.

Expand a product, then evaluate

Simplify $3x(4x - 3) + 3$ and find its value at $x = 1$ and $x = \tfrac{1}{2}$.

Expanding with the distributive law:

$3x(4x - 3) + 3 = 12x^2 - 9x + 3.$

At $x = 1$ (without expanding). Substitute into the original form:

$3(1)\big(4(1) - 3\big) + 3 = 3(4 - 3) + 3 = 3 + 3 = 6.$

At $x = 1$ (using the expansion). The expanded form must agree:

$12(1)^2 - 9(1) + 3 = 12 - 9 + 3 = 6.$

At $x = \tfrac{1}{2}$. Substitute into the expansion:

$12\left(\tfrac{1}{2}\right)^2 - 9\left(\tfrac{1}{2}\right) + 3 = 12 \cdot \tfrac{1}{4} - \tfrac{9}{2} + 3 = 3 - \tfrac{9}{2} + 3 = 6 - \tfrac{9}{2} = \tfrac{3}{2}.$

Simplify and evaluate $a(a^2 + a + 1) + 5$

Expanding:

$a(a^2 + a + 1) + 5 = a^3 + a^2 + a + 5.$

At $a = 0$: $\;0 + 0 + 0 + 5 = 5.$

At $a = 1$: $\;1 + 1 + 1 + 5 = 8.$

At $a = -1$: odd powers of $-1$ give $-1$ and even powers give $+1$, so

$(-1)^3 + (-1)^2 + (-1) + 5 = -1 + 1 - 1 + 5 = 4.$

Add three products in $p$, $q$, $r$

Add $p(p - q)$, $q(q - r)$ and $r(r - p)$:

$p(p - q) + q(q - r) + r(r - p) = p^2 - pq + q^2 - qr + r^2 - rp.$

Collecting terms:

$= p^2 + q^2 + r^2 - pq - qr - rp.$

A second sum

Add $2x(z - x - y)$ and $2y(z - y - x)$:

$2x(z - x - y) = 2xz - 2x^2 - 2xy,$ $2y(z - y - x) = 2yz - 2y^2 - 2xy.$

Adding and grouping like terms:

$= 2xz + 2yz - 2x^2 - 2y^2 - 4xy.$

Subtract products: the column method

Subtract $3l(l - 4m + 5n)$ from $4l(10n - 3m + 2l)$.

The two products.

$3l(l - 4m + 5n) = 3l^2 - 12lm + 15ln,$ $4l(10n - 3m + 2l) = 40ln - 12lm + 8l^2.$

Subtract by adding the additive inverse: change the subtraction to addition and flip the sign of every term being subtracted, so $-(3l^2 - 12lm + 15ln)$ becomes $-3l^2 + 12lm - 15ln$. Lining up like terms in columns:

8l^2 - 12lm + 40ln \\ -3l^2 + 12lm - 15ln \end{aligned}$$ Adding column by column: $8l^2 - 3l^2 = 5l^2$, $\;-12lm + 12lm = 0$, $\;40ln - 15ln = 25ln$. $$\text{Result} = 5l^2 + 25ln.$$ ### Subtract products: the row method Subtract $\;3a(a + b + c) - 2b(a - b + c)\;$ from $\;4c(-a + b + c)$. **First expression (the one being subtracted).** $$3a(a + b + c) - 2b(a - b + c) = 3a^2 + 3ab + 3ac - 2ab + 2b^2 - 2bc.$$ $$= 3a^2 + ab + 3ac + 2b^2 - 2bc.$$ **Second expression.** $$4c(-a + b + c) = -4ac + 4bc + 4c^2.$$ **Subtract** by flipping the signs of the first expression and adding: $$(-4ac + 4bc + 4c^2) + (-3a^2 - ab - 3ac - 2b^2 + 2bc).$$ Collecting like terms in a row: $-4ac - 3ac = -7ac$, $\;4bc + 2bc = 6bc$, and the rest carry through. $$\text{Result} = -3a^2 - 2b^2 + 4c^2 - ab + 6bc - 7ac.$$ ### Key takeaways - Expand a product with the distributive law; an expression evaluates to the same value whether you substitute before or after expanding. - To subtract one expression from another, add its additive inverse: change the sign of every term and then add. - Use the column method when terms line up neatly, and the row method when the expressions have several unlike terms; both just keep like terms together.