EQN-03-05

Simplifying Expressions & Factoring

Combine like terms, expand products, and use the three factoring patterns the GAT-E rewards.

~6 min read•Difficulty 3/5

Only terms with the same variable part combine $.3 x + 5 x = 8 x,$ but $3 x + 5 x^{2}$ stays as it is.

a (b + c) = ab + ac

(x + 2) (x + 3) = x^{2} + 5 x + 6 (First, Outer, Inner, Last)

Common factor : ab + ac = a (b + c)

Difference of squares : a^{2} - b^{2} = (a - b) (a + b)

Trinomial : x^{2} + (p + q) x + pq = (x + p) (x + q)

It turns "hard" arithmetic into easy products and reveals shortcuts.

4 9^{2} - 1 = (49 - 1) (49 + 1) = 48 \times 50 = 2400 (no long multiplication)

To find the value of an expression, factor first when you can, then substitute — it usually avoids large numbers.

Worked examples

Example 1 (difficulty 2/5)

Factor

: x^{2} - 9.

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Expand and simplify

: (x + 4) (x - 2) .

◎ Predict the answer first:

Spot $a^{2} - b^{2}$ instantly — it is the most-rewarded factoring pattern on the test.
For a trinomial $x^{2} +$ bx $+ c,$ find two numbers that multiply to c and add to b.
Factoring large "difference of squares" numbers beats long multiplication.

Treating

(a + b)^{2}

a^{2} + b^{2}

Why: It looks like the square distributes.

Fix:

(a + b)^{2} = a^{2} + 2

+ b^{2}

— never forget the middle term 2ab.

Combining unlike terms

(3 x + 4 = 7 x)

Why: The numbers invite addition.

Fix: Only terms with identical variable parts combine; 3x and 4 do not.