Polynomial Roots and Discriminant: Quick Lesson and UPCAT Practice

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Polynomial Roots and Discriminant

Connect zeros, factors, and graphs, then use the discriminant to classify roots without unnecessary solving.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Polynomial Roots and Discriminant

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QUICK REVIEW

Roots, zeros, and factors describe the same event

A root or zero is an input that makes a polynomial equal zero. If P(r)=0, then r is a root and (x−r) is a factor. On a graph, real roots are the x-values where the graph meets the x-axis.

See roots and factors in action

Find the roots of P(x)=x²−5x+6.

Factor: P(x)=(x−2)(x−3).

Set each factor equal to zero: x−2=0 or x−3=0. Therefore, the roots are x=2 and x=3.

Check: P(2)=0 and P(3)=0, so the factors and roots agree.

When a quadratic does not factor easily—or when the question asks only how many real roots it has—use the discriminant: Δ=b²−4ac.

See the discriminant in action

For x²−6x+9=0, a=1, b=−6, and c=9.

Δ=b²−4acΔ=(−6)²−4(1)(9)Δ=36−36Δ=0

Because Δ=0, the equation has one repeated real root. In fact, x²−6x+9=(x−3)².

Factor theorem

P(r)=0 exactly when (x−r) is a factor.

Discriminant

Δ > 0 → two distinct real rootsΔ = 0 → one repeated real rootΔ < 0 → no real roots

DO IT FAST

Classify before you solve

If a question asks only about the kind or number of roots, calculate Δ. Do not use the full quadratic formula unless actual root values are required.

Why it works

The quadratic formula contains √Δ. Its sign determines whether the square root is positive, zero, or not real.

WORKED EXAMPLES

Five forms you should recognize

1. Zero to factor

If 3 is a zero, then (x−3) is a factor.

2. Factor to zero

(x+2)(x−5)=0 gives roots −2 and 5.

3. Two real roots

x²−5x+4=0Δ=b²−4acΔ=(−5)²−4(1)(4)Δ=25−16=9Since Δ > 0, there are two distinct real roots.

4. Repeated root

4x²+4x+1=0Δ=b²−4acΔ=4²−4(4)(1)Δ=16−16=0Since Δ = 0, there is one repeated real root.

5. Known cubic root

If a cubic has factor (x−1), divide it out and analyze the remaining quadratic.

COMMON TRAPS

Check before you commit

• Wrong factor sign: root 3 corresponds to x−3.
• Δ is not the root: it classifies roots unless used in the full formula.
• Forgetting a: Δ=b²−4ac includes all coefficients.
• Missing multiplicity: a squared factor gives a repeated root.
• Assuming real: a negative discriminant means no real roots.
• Remainder confusion: division by x−r gives remainder P(r).

FIVE-FORM SKILL CHECK

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Core Practice

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QUICK ANSWERS

Polynomial Roots and Discriminant FAQ

Are roots and zeros different?

In this context they refer to input values that make the polynomial equal zero.

What does Δ=0 mean graphically?

The parabola touches the x-axis at one repeated root.

Does Δ<0 mean the polynomial has no roots at all?

It has no real roots, though complex roots exist beyond the usual UPCAT scope.

How does the remainder theorem help?

The remainder after division by x−r is P(r); a zero remainder confirms a factor.

RELATED COMPETENCIES

Continue your mathematics review.

FactoringQuadratic EquationsPolynomial DivisionRemainder TheoremQuadratic Graphs

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