Pre-Calculus · Factoring
Pre-Calc Diagnostic: Factor Each Expression (Stewart Problem 4)
Six factoring techniques from Stewart's Pre-Calculus Diagnostic Test — difference of squares, reverse-FOIL on a non-monic trinomial, factoring by grouping, sum of cubes, factoring with negative fractional exponents, and two-variable factoring. The six patterns every pre-calc and AP Calculus student should own, taught the way we teach them at SOMATH on the Upper West Side.
Direct answer: Problem 4 of Stewart's Pre-Calculus diagnostic asks you to factor six expressions using six different techniques. The answers are (a) (2x − 5)(2x + 5), (b) (2x − 3)(x + 4), (c) (x − 2)(x + 2)(x − 3), (d) x(x + 3)(x2 − 3x + 9), (e) 3x−1/2(x − 1)(x − 2) = 3(x − 1)(x − 2)x, and (f) xy(x − 2)(x + 2). The hardest of the six is (e), where you factor out a negative fractional exponent — pull out the lowest power, not the smallest number. Below is the full walkthrough we use at SOMATH on the Upper West Side.
The six factoring patterns every pre-calc student should own
Before we touch the algebra, here's the menu. When you see one of these structures, you should recognize the move immediately:
- Difference of squares — a2 − b2 = (a − b)(a + b). Two terms, both perfect squares, minus sign.
- Sum/difference of cubes — a3 + b3 = (a + b)(a2 − ab + b2), a3 − b3 = (a − b)(a2 + ab + b2).
- Trinomial reverse-FOIL — for ax2 + bx + c, find two numbers whose product is ac and sum is b.
- Factoring by grouping — four terms, pair them, pull common factors, look for a matching binomial.
- GCF first, always — including negative or fractional exponents. Pull out the LOWEST power.
Every part of Problem 4 uses one (or two) of these. Now let's work them.
(a) Difference of squares: 4x2 − 25
Both terms are perfect squares: 4x2 = (2x)2 and 25 = 52. The minus sign confirms difference of squares. Apply the pattern directly:
4x2 − 25 = (2x)2 − 52 = (2x − 5)(2x + 5)
The order of the two factors doesn't matter — (2x + 5)(2x − 5) is the same product. Either way is correct.
(b) Trinomial by reverse-FOIL: 2x2 + 5x − 12
This is a non-monic trinomial — the leading coefficient is 2, not 1, so we can't just look for "two numbers that multiply to −12 and add to 5." We need the AC method. Multiply a · c = 2 · (−12) = −24. Now find two numbers whose product is −24 and sum is +5. Try pairs: 8 and −3 works (8 · (−3) = −24, 8 + (−3) = 5).
Split the middle term and group:
2x2 + 5x − 12 = 2x2 + 8x − 3x − 12 = 2x(x + 4) − 3(x + 4) = (2x − 3)(x + 4)
Check by FOIL: (2x − 3)(x + 4) = 2x2 + 8x − 3x − 12 = 2x2 + 5x − 12. Done.
(c) Factoring by grouping: x3 − 3x2 − 4x + 12
Four terms is the visual cue for grouping. Pair them in order and pull the GCF from each pair:
x3 − 3x2 − 4x + 12 = x2(x − 3) − 4(x − 3)
Watch the sign in the second pair: we factored out −4, not +4, so that (x − 3) matches the first group. Now the binomial (x − 3) is common — pull it out:
= (x − 3)(x2 − 4)
But x2 − 4 is itself a difference of squares: (x − 2)(x + 2). The fully-factored answer is:
(x − 2)(x + 2)(x − 3)
This is a common student trap — they stop at (x − 3)(x2 − 4). Always look at the remaining factors to see if anything else will factor.
(d) Sum of cubes after common factor: x4 + 27x
The first move on EVERY factoring problem: look for a GCF. Here both terms share an x:
x4 + 27x = x(x3 + 27)
Now x3 + 27 = x3 + 33 is a sum of cubes. Apply the formula a3 + b3 = (a + b)(a2 − ab + b2) with a = x and b = 3:
x3 + 27 = (x + 3)(x2 − 3x + 9)
Putting it together:
x(x + 3)(x2 − 3x + 9)
The quadratic factor x2 − 3x + 9 is irreducible over the reals — its discriminant (−3)2 − 4(1)(9) = 9 − 36 = −27 is negative, so there's no further factoring with real numbers.
(e) Factoring with negative fractional exponents: 3x3/2 − 9x1/2 + 6x−1/2
This is the one that trips up most pre-calc students. The instinct is to factor out 3x1/2 because that "looks like" the smallest power. Wrong. The rule for factoring with exponents — fractional, negative, or otherwise — is to pull out the LOWEST exponent. Compare the three exponents: 3/2, 1/2, −1/2. The lowest is −1/2.
The numerical GCF is 3. So we factor out 3x−1/2. When you pull x−1/2 out of each term, every remaining exponent gains +1/2 (because x−1/2 · x2 = x3/2, so the partner inside must be x2):
3x3/2 − 9x1/2 + 6x−1/2 = 3x−1/2(x2 − 3x + 2)
Now the quadratic in parentheses is a friendly trinomial: two numbers multiplying to +2, summing to −3, are −1 and −2. So:
x2 − 3x + 2 = (x − 1)(x − 2)
Final factored form:
3x−1/2(x − 1)(x − 2)
Stewart's solution key writes this with the negative exponent moved to the denominator using x−1/2 = 1/x:
= 3(x − 1)(x − 2)x
Both forms are correct. The "no negative exponent" version is preferred for a final written answer; the negative-exponent version is preferred when you're about to take a derivative.
(f) Two-variable factoring + difference of squares: x3y − 4xy
GCF first. Both terms share xy:
x3y − 4xy = xy(x2 − 4)
And x2 − 4 is a difference of squares — x2 − 22 = (x − 2)(x + 2):
xy(x − 2)(x + 2)
Same principle as part (c): pull the GCF, then check whether the remaining polynomial factors further. In this case it does, so the final form has three factors.
The pattern behind all six
Every Stewart Problem 4 part follows the same diagnostic order: (1) GCF first, including negative or fractional exponents, (2) count the terms — two suggests difference of squares or sum/difference of cubes, three suggests trinomial reverse-FOIL, four suggests grouping, (3) factor each remaining piece until nothing more will factor. Students who memorize the order — GCF, count, recognize, repeat — never get stuck.
The negative-exponent move in part (e) is the highest-yield insight here. It comes back in AP Calculus when you simplify derivatives of products that contain x, and it appears on the SAT Math 2 subject content. If your student can do (e) cleanly, they're solid on the whole problem.
How SOMATH teaches factoring on the Upper West Side
At SOMATH (226 W 79th St, UWS), we teach factoring as a decision tree, not a memorized list. Every student gets a one-page flowchart in week one: "Look at the expression — is there a GCF? Pull it. How many terms now? Two → squares or cubes. Three → reverse-FOIL. Four → grouping." Drill that order through Stewart's diagnostic set, then on word-problem applications (volume, area, projectile motion), and the technique becomes automatic before AP Calc.
If your high schooler is heading into pre-calc, AP Calc AB/BC, or SAT Math 2, the diagnostic Problem 4 is a great two-day session: one day on (a)–(d), one on (e)–(f) plus a worksheet of mixed-strategy problems. Most students walk in able to do (a) and (b) and walk out able to do all six.
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FAQ
When should I factor by grouping vs trial-and-error?
Group when you have four terms — pull a common factor out of the first pair and out of the second pair, then look for a matching binomial. Use trial-and-error (reverse-FOIL) when you have a trinomial ax2 + bx + c. If grouping doesn't yield a common binomial, the expression may not factor over the integers.
Why do you pull out a NEGATIVE exponent when factoring 3x3/2 − 9x1/2 + 6x−1/2?
The rule is: factor out the LOWEST power of x, including negative ones. The lowest power here is x−1/2. When you pull it out, every remaining exponent gains +1/2, so the inside becomes integer powers and the expression is easier to factor further.
How do I recognize a sum of cubes inside a polynomial that doesn't look like one?
Always pull common factors first. In x4 + 27x, factoring out x reveals x(x3 + 27) — and x3 + 27 = x3 + 33 is a clean sum of cubes. The general formula: a3 + b3 = (a + b)(a2 − ab + b2).
Is the difference of squares only for two-term expressions?
Yes — difference of squares means a2 − b2 = (a − b)(a + b). It applies whenever you have exactly two perfect squares separated by a minus sign. In part (f), once you factor xy out of x3y − 4xy you get xy(x2 − 4), which IS a difference of squares: xy(x − 2)(x + 2).
Do students need to memorize the sum-of-cubes formula for pre-calc?
Yes. The two cubic identities — a3 + b3 = (a + b)(a2 − ab + b2) and a3 − b3 = (a − b)(a2 + ab + b2) — appear throughout pre-calc, AP Calculus AB/BC, and the SAT Math 2 subject content. They're worth permanent memory, along with the difference of squares and perfect-square trinomials.