A polynomial can be read in several ways at once: as an expression to manipulate, a product of factors, an equation with roots and a function with graph features. The strongest solutions connect these views instead of treating factorisation as a mechanical end point.
1. What is a polynomial?
A polynomial in x is built from nonnegative integer powers of x with constant coefficients. For example, 3x⁴−2x²+x−7 is a polynomial of degree 4. Expressions such as 1/x, √x and x^(3/2) are not polynomials in x.
2. Degree predicts the maximum number of roots
A nonzero polynomial of degree n has at most n distinct real roots. Over the complex numbers, counting multiplicity, it has n roots. At school level, this gives a useful structural check: a cubic cannot have four distinct real roots.
3. Factors and roots are connected
If P(a)=0, then x−a is a factor of P(x). Conversely, if x−a is a factor, then a is a root of P(x)=0.
For P(x)=(x−2)(x+1)(x−5), the roots are 2, −1 and 5. Each factor becomes zero at its corresponding root.
4. Multiplicity records repeated factors
In P(x)=(x−3)²(x+2), x=3 is a root of multiplicity 2 and x=−2 has multiplicity 1.
Repeated roots matter because the polynomial contains the same factor more than once. They also affect graph behaviour near the horizontal axis.
5. Leading coefficient controls scale and end behaviour
Two polynomials can have the same roots but different vertical scales. P(x)=(x−1)(x+2) and Q(x)=5(x−1)(x+2) have identical roots but different outputs elsewhere.
The highest-degree term dominates for large |x|. Thus a positive even-degree leading term rises at both ends, while a negative even-degree leading term falls at both ends. Odd degree gives opposite end directions.
6. Worked example: build structure from roots
A cubic has roots 1, 1 and −4 and leading coefficient 2. Therefore P(x)=2(x−1)²(x+4).
The repeated root is represented by the squared factor. Expanding is unnecessary unless the question asks for standard polynomial form or a coefficient.
7. Worked example: find an unknown coefficient
Let P(x)=x³+ax²−5x+3 and suppose x−1 is a factor. Then P(1)=0, so 1+a−5+3=0. Hence a=1.
The factor condition becomes a substitution condition. This is often faster than attempting full factorisation first.
8. Coefficient relationships provide checks
If a monic cubic has roots r, s and t, then (x−r)(x−s)(x−t)=x³−(r+s+t)x²+(rs+rt+st)x−rst.
For roots 1, 2 and 3, the polynomial is x³−6x²+11x−6. The constant term −6 agrees with −(1×2×3).
9. Common mistakes
- Writing factor x+3 for root 3 instead of x−3.
- Forgetting the leading multiplier when roots are given.
- Treating a repeated root as two different root values.
- Expanding immediately when factor form already answers the question.
10. Practice
- State the degree of 4x⁵−x²+7.
- State the roots and multiplicities of (x+2)³(x−5).
- Construct a monic quadratic with roots 3 and −4.
- Find k if x−2 is a factor of x³+kx−10.
- Construct a cubic with leading coefficient −2 and roots 0, 1 and 1.
11. Answers
- 5.
- −2 with multiplicity 3; 5 with multiplicity 1.
- (x−3)(x+4)=x²+x−12.
- P(2)=8+2k−10=0, so k=1.
- −2x(x−1)².
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