6.2 Exponents, Factoring, and Systems of Equations

Why This Matters for the PUPCET

Exponents, factoring, and systems of equations form the middle band of the PUPCET Mathematics portion, which carries roughly 21% of the entire test. These items reward pattern recognition more than long computation: a student who instantly sees that x² − 49 is (x + 7)(x − 7) finishes in seconds, while one who tries to factor by trial and error burns a minute. Because the exam gives you a little over a minute per item, this section is where disciplined memorization converts directly into points.

The Laws of Exponents

An exponent tells you how many times a base is multiplied by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The PUPCET assumes you can apply all six laws below fluently, including with negative and zero exponents.

Worked Example 1: Simplify with Exponent Laws

Simplify: (2x³y²)(3x²y⁵)

Multiply the coefficients: 2 × 3 = 6. For x, add the exponents: 3 + 2 = 5. For y: 2 + 5 = 7. Answer: 6x⁵y⁷. The trap here is multiplying the exponents (getting x⁶y¹⁰) — remember that multiplication of powers means ADD the exponents. You multiply exponents only when a power is raised to another power, as in (x³)⁴ = x¹².

Worked Example 2: Negative and Zero Exponents

Evaluate: 5⁰ + 3⁻²

Any nonzero base to the zero power is 1, so 5⁰ = 1. For the negative exponent, take the reciprocal: 3⁻² = 1/3² = 1/9. So the sum is 1 + 1/9 = 10/9 (or 1 1/9). A common wrong answer treats 3⁻² as −9 or −6. A negative exponent never makes the value negative — it makes it a fraction.

Special Products You Must Recognize Instantly

Three expansion patterns appear over and over on Philippine entrance exams. Memorize them in both directions — expanding and factoring.

Worked Example 3: Square of a Binomial

Expand: (x + 6)²

Using the pattern: x² + 2(x)(6) + 6² = x² + 12x + 36. The single most-tested trap in this whole topic is writing x² + 36 and dropping the middle term. The PUPCET will absolutely offer x² + 36 as a decoy. You can always verify by writing the binomial twice and using FOIL (First, Outer, Inner, Last): (x + 6)(x + 6) = x² + 6x + 6x + 36.

Worked Example 4: Difference of Squares for Fast Arithmetic

Compute 51 × 49 without a calculator.

Notice 51 = 50 + 1 and 49 = 50 − 1, so the product is (50 + 1)(50 − 1) = 50² − 1² = 2500 − 1 = 2499. This trick converts ugly multiplication into a one-line subtraction and occasionally appears directly as a PUPCET item.

Factoring Simple Trinomials

Factoring reverses expansion: it rewrites a polynomial as a product of simpler factors. For a trinomial of the form x² + bx + c, find two numbers that multiply to c and add to b.

Worked Example 5: Factor x² + 7x + 12

Find two numbers whose product is 12 and whose sum is 7. Pairs of factors of 12: (1, 12), (2, 6), (3, 4). Check sums: 1 + 12 = 13, 2 + 6 = 8, 3 + 4 = 7. The pair is 3 and 4, so x² + 7x + 12 = (x + 3)(x + 4). Always verify by expanding back: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12. ✓

Worked Example 6: Factor with Negative Signs

Factor: x² − 2x − 15

You need two numbers with product −15 and sum −2. Since the product is negative, the numbers have opposite signs. Pairs multiplying to −15: (−1, 15), (1, −15), (−3, 5), (3, −5). Their sums: 14, −14, 2, −2. The pair is 3 and −5, so x² − 2x − 15 = (x + 3)(x − 5).

Sign rules worth memorizing: if c is positive, both numbers share the sign of b; if c is negative, the numbers have opposite signs and the larger absolute value takes the sign of b.

Solving Systems of Two Equations

A system of equations asks for the one (x, y) pair that satisfies two equations at once. Graphically it is the point where two lines cross. The PUPCET expects both algebraic methods.

Method 1: Substitution

Use substitution when one equation already has a variable isolated.

Solve: y = 2x + 1 and 3x + y = 16

Step 1. Substitute the expression for y into the second equation: 3x + (2x + 1) = 16. Step 2. Simplify: 5x + 1 = 16, so 5x = 15 and x = 3. Step 3. Back-substitute: y = 2(3) + 1 = 7. Answer: (3, 7). Check in the second equation: 3(3) + 7 = 16. ✓

Method 2: Elimination

Use elimination when matching or opposite coefficients let you cancel a variable.

Solve: 2x + 3y = 12 and 2x − y = 4

Step 1. Subtract the second equation from the first to eliminate x: (2x − 2x) + (3y − (−y)) = 12 − 4, giving 4y = 8, so y = 2. Step 2. Substitute into the second equation: 2x − 2 = 4, so 2x = 6 and x = 3. Answer: (3, 2). Check in the first equation: 2(3) + 3(2) = 6 + 6 = 12. ✓

If neither variable cancels directly, multiply one whole equation by a constant first. For 3x + 2y = 16 and x + y = 6, multiply the second equation by 2 to get 2x + 2y = 12, then subtract to find x = 4 and y = 2.

Shortcut Exam Techniques

• Plug the options in. On a multiple-choice system, substitute each candidate (x, y) into the easier equation first. Usually only one option survives, and you never do full algebra.
• Add the equations strategically. If asked for x + y rather than x and y separately, try adding the two equations directly — the sum sometimes appears without solving for each variable.
• Recognize special products in disguise. Items like 999² look brutal until you write (1000 − 1)² = 1,000,000 − 2,000 + 1 = 998,001.
• Factor before solving quadratics. To solve x² − 5x + 6 = 0, factor to (x − 2)(x − 3) = 0 and read off x = 2 or x = 3 via the zero-product property: if a product is zero, at least one factor is zero.

Common Traps

• Dropping the middle term of (a ± b)².
• Multiplying exponents when you should add them (x² · x³ = x⁵, not x⁶).
• Thinking x⁰ = 0. Every nonzero base to the zero power equals 1.
• Forgetting that a negative exponent produces a reciprocal, not a negative number.
• In elimination, subtracting a negative term incorrectly: 3y − (−y) = 4y, not 2y.

Section Takeaway

Add exponents when multiplying like bases, subtract when dividing, multiply when raising a power to a power. Burn the three special-product patterns into memory in both directions. Factor trinomials by hunting for a product-sum pair. Choose substitution when a variable is isolated and elimination when coefficients line up — and when stuck, plug the answer choices back in.