Polynomials
Senior 2 · Mathematics
Polynomials
Lesson 1 of 5
22 min
Throw a ball straight up at 20 m per second and its height after t seconds is about 20t − 5t2 metres. A farmer's rectangular plot that is x metres wide and (x + 3) metres long has an area of x2 + 3x square metres. Expressions like these, built from a variable, whole-number powers and numbers, are called polynomials.
Polynomials are the everyday language of algebra. Engineers use them to model bridges and roads, economists to model costs, and computers to draw smooth curves. This unit teaches you to name them, calculate with them and factorise them.
Look at P(x) = 3x4 − 2x2 + x − 7.
A polynomial in x is a sum of terms of the form axn, where each coefficient a is a real number and each index n is a whole number (0, 1, 2, …). So 2/x (which is 2x−1) and √x (which is x12) are not polynomials. We usually write a polynomial in descending powers, as above.
| By number of terms | Example | By degree | Example |
|---|---|---|---|
| Monomial (1 term) | −5x3 | Constant (degree 0) | 8 |
| Binomial (2 terms) | x + 4 | Linear (degree 1) | 2x − 1 |
| Trinomial (3 terms) | x2 − 5x + 6 | Quadratic (degree 2) | x2 + 3x |
| Polynomial (4 or more) | x3 + x2 − x + 1 | Cubic (degree 3) | 4x3 − x |
Like terms have the same variable raised to the same power (3x2 and −7x2). To add polynomials, collect like terms.
Worked example 1
(4x3 − 3x2 + 5) + (x3 + 6x2 − 2x − 9)
x3: 4 + 1 = 5. x2: −3 + 6 = 3. x: −2. Constants: 5 − 9 = −4. Answer: 5x3 + 3x2 − 2x − 4.
To subtract, change the sign of every term in the second bracket, then add.
Worked example 2
(5x2 − 3x + 2) − (2x2 + 4x − 6) = 5x2 − 3x + 2 − 2x2 − 4x + 6 = 3x2 − 7x + 8.
The numerical value of P(x) at x = a, written P(a), is found by replacing every x by a.
Worked example 3
P(x) = 2x3 − 5x + 4. Then P(2) = 2(8) − 5(2) + 4 = 16 − 10 + 4 = 10, and P(−1) = 2(−1) − 5(−1) + 4 = −2 + 5 + 4 = 7.
For the ball above, h(t) = 20t − 5t2 gives h(2) = 40 − 20 = 20. After 2 seconds the ball is 20 m high.
Two polynomials are equal (identical) when they have the same degree and the coefficients of like powers are equal.
Worked example 4
Find a, b and c if ax2 + (b − 1)x + 6 = 5x2 + 3x + (c + 2) for every x.
Compare x2: a = 5. Compare x: b − 1 = 3, so b = 4. Compare constants: 6 = c + 2, so c = 4.
Tip
Exam technique. When you subtract a bracket, the minus sign changes every sign inside it, not just the first one. Write the step out in full: −(2x2 + 4x − 6) = −2x2 − 4x + 6. Most lost marks in this topic come from skipping that line.
State the degree, leading coefficient and constant term of 6 − x + 4x3.
Solution: In descending order it is 4x3 − x + 6. Degree 3 (a cubic), leading coefficient 4, constant term 6.
Simplify (2x2 + 5x − 3) + (x2 − 7x + 8).
Solution: 3x2 − 2x + 5.
Simplify (3x2 − 4x + 1) − (x2 − 6x + 5).
Solution: 3x2 − 4x + 1 − x2 + 6x − 5 = 2x2 + 2x − 4.
If Q(x) = x2 − 3x + 2, find Q(4) and Q(1).
Solution: Q(4) = 16 − 12 + 2 = 6. Q(1) = 1 − 3 + 2 = 0.
Find m and n if (m + 2)x2 − 4x + n = 7x2 − 4x − 1.
Solution: m + 2 = 7, so m = 5; and n = −1.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
