Degree, type and terms of a polynomial
A polynomial is like a recipe: each term is an ingredient with an amount, and the highest power tells you its "size".
A polynomial in x has terms with whole-number powers of x. Its degree is the highest power. Degree 1 is linear, 2 is quadratic, 3 is cubic.
In the figure: 3x² − 5x + 2 has three terms and degree 2, so it is quadratic.
Rule 1: Name the degree — The biggest power of x decides.
- Easy: 2x − 7: degree 1, linear.
- Medium: x³ − x + 4: degree 3, cubic.
- Exam-type: x⁴ − 1: degree 4, a biquadratic.
Rule 2: What is not a polynomial — Powers must be whole numbers.
- Easy: x + 1/x has x⁻¹: not a polynomial.
- Medium: √x + 2 has x^(1/2): not a polynomial.
- Exam-type: 7 is a constant polynomial (degree 0).
A polynomial of degree n has at most n terms with a different power each, plus a constant.
Taking the number of terms as the degree.
"Degree = highest power."
Practice questions
- 1What is the degree of the polynomial −4x − 9x²?
- 2A polynomial of degree 5 can have at most how many zeroes?
- 3The polynomial 5x² − x − 7 is a ___ polynomial.
- 4The polynomial 5x − 9 is a ___ polynomial.
- 5The polynomial −5x² + 7x is a ___ polynomial.
- 6A polynomial of degree 7 can have at most how many zeroes?
- 7If p(x) = 5x³ − 5x² − 3x − 8 and q(x) = −5x³ + 8x² − 5x + 2, what is the degree of p(x) + q(x)?
- 8If p(x) = 6x³ + x² + 8x − 7 and q(x) = −6x³ + 5x² − 7x + 1, what is the degree of p(x) + q(x)?
Show answers (try them yourself first)
- 12
- 25
- 3quadratic
- 4linear
- 5quadratic
- 67
- 72
- 82
Value of a polynomial and its zero
Put a number into a formula and you get a result. The inputs that give 0 are special: they are the zeroes.
The value of p(x) at x = k is p(k). If p(k) = 0, then k is a zero of the polynomial.
Formula: k is a zero ⇔ p(k) = 0
In the figure: For x² − 3x + 2: p(1) = 1 − 3 + 2 = 0 and p(2) = 4 − 6 + 2 = 0, but p(3) = 2.
Rule 1: Evaluate — Substitute and simplify.
- Easy: p(x) = x² + x − 6: p(−1) = 1 − 1 − 6 = −6.
- Medium: p(x) = 2x − 6: p(3) = 0.
- Exam-type: p(x) = x³ − x: p(2) = 8 − 2 = 6.
Rule 2: Test a zero — Zero value means zero of the polynomial.
- Easy: x² − 4: p(2) = 0 and p(−2) = 0, both zeroes.
- Medium: x² + 1: p(1) = 2 ≠ 0.
- Exam-type: A zero can itself be 0: p(x) = x² has zero 0.
A linear polynomial ax + b has exactly one zero, −b/a.
Confusing the zero k with the value p(k).
"Zero in, zero out ⇒ zero of p."
Practice questions
- 1If p(x) = −4x³ − 2x² − 7x + 3, find p(−1).
- 2If p(x) = −6x² + 4x + 2, find p(−3).
- 3If p(x) = −x² − 2x + 5, find p(2).
- 4If p(x) = 5x² + 4x − 4, find p(−1).
- 5If x = −4 is a zero of the polynomial x² + kx − 8, find the value of k.
- 6If x = 4 is a zero of the polynomial kx² − 3x + 60, find the value of k.
- 7If x = 4 is a zero of the polynomial x³ − 3x² + kx + 20, find the value of k.
- 8If x = 4 is a zero of the polynomial x² + kx − 20, find the value of k.
Show answers (try them yourself first)
- 112
- 2−64
- 3−3
- 4−3
- 52
- 6−3
- 7−9
- 81
Linear polynomial: zero and graph
A ramp has one place where it touches the ground. A straight line y = ax + b crosses the x-axis exactly once.
The graph of y = ax + b (a ≠ 0) is a straight line. Its zero is the x-coordinate of the point where it cuts the x-axis, and equals −b/a.
Formula: zero of ax + b is −b/a
In the figure: y = 2x − 4 cuts the x-axis at x = 2, so its zero is 2.
Rule 1: Read the zero — Look at where the line meets the x-axis.
- Easy: y = x − 3: zero 3.
- Medium: y = 2x + 6: zero −3.
- Exam-type: y = 4x: passes through the origin, zero 0.
Rule 2: Use −b/a — No graph needed for a linear polynomial.
- Easy: 3x − 9: zero 9/3 = 3.
- Medium: 5x + 10: zero −10/5 = −2.
- Exam-type: 7x + 5: zero −5/7.
A line parallel to the x-axis has no zero; the line y = 0 itself has all numbers as zeroes.
Writing the zero as b/a instead of −b/a.
"Line crosses once: zero = −b/a."
Practice questions
- 1The zero of a polynomial p(x) is the ___ of the point where the graph meets the x-axis.
- 2Find the zero of the linear polynomial −3x + 10.
- 3Find the zero of the linear polynomial 3x − 2.
- 4Find the zero of the linear polynomial 4x + 12.
- 5Find the zero of the linear polynomial −4x + 8.
- 6Find the zero of the linear polynomial −2x − 5.
- 7Find the zero of the linear polynomial −5x + 12.
- 8Find the zero of the linear polynomial −7x − 8.
Show answers (try them yourself first)
- 1x-coordinate
- 210/3
- 32/3
- 4−3
- 52
- 6−5/2
- 712/5
- 8−8/7
Quadratic polynomial: parabola and number of zeroes
A thrown ball follows a curve called a parabola. It can cross the ground twice, just touch it, or never reach it.
The graph of y = ax² + bx + c is a parabola: it opens upward if a > 0, downward if a < 0. A quadratic has at most 2 zeroes: it cuts the x-axis at 2, 1 or 0 points.
Formula: a > 0 ∪ opens up; a < 0 ∩ opens down
In the figure: y = x² − 4 opens upward and cuts the x-axis at −2 and 2, so it has two zeroes.
Quadratic polynomial: parabola and number of zeroes
In the app you can move this figure with your finger.
Open in the appRule 1: Count the zeroes — Count the points where the parabola meets the x-axis.
- Easy: x² − 4: cuts twice, zeroes ±2.
- Medium: x²: touches at 0 only, one zero.
- Exam-type: x² + 1: stays above the axis, no zero.
Rule 2: Direction of opening — The sign of a decides up or down.
- Easy: x² − 2x: a = 1 > 0, opens up.
- Medium: −x² + 1: a = −1 < 0, opens down.
- Exam-type: −x² + 1 has zeroes ±1.
A quadratic can never have more than 2 zeroes.
Thinking the lowest point of the parabola is its zero.
"Up if a is plus; zeroes are the x-axis hits."
Practice questions
- 1The parabola y = 3x² + 5x − 6 opens ___.
- 2The parabola y = 6x² − 6x − 5 opens ___.
- 3The parabola y = x² − 6x + 1 opens ___.
- 4How many times does the graph of y = 4x² − 6x − 4 cut or touch the x-axis?
- 5How many times does the graph of y = 4x² − x − 5 cut or touch the x-axis?
- 6A parabola opening upwards lies completely above the x-axis. How many zeroes does the polynomial have?
- 7For which value of k does the graph of y = x² + 6x + k touch the x-axis at exactly one point?
- 8For which value of k does the graph of y = x² + 14x + k touch the x-axis at exactly one point?
Show answers (try them yourself first)
- 1upwards
- 2upwards
- 3upwards
- 42
- 52
- 60
- 79
- 849
Cubic and higher polynomials
A rollercoaster that goes up, down and up again can touch the ground level up to three times. A cubic works the same way.
A polynomial of degree n has at most n zeroes. A cubic cuts the x-axis at 1, 2 or 3 points. Its graph is an S-like curve.
Formula: zeroes ≤ degree
In the figure: y = x³ − x cuts the x-axis three times, at −1, 0 and 1.
Rule 1: How many zeroes? — Use the graph; the count is at most the degree.
- Easy: x³ − x: 3 zeroes.
- Medium: x³: touches at 0, one zero.
- Exam-type: x³ + x: one zero (0) only.
Rule 2: Check by factorising — Zeroes are where each factor is zero.
- Easy: x³ − x = x(x − 1)(x + 1): zeroes 0, 1, −1.
- Medium: (x − 2)(x + 3)x: zeroes 2, −3, 0.
- Exam-type: Degree n ⇒ at most n zeroes.
Every cubic has at least one zero because its graph must cross the x-axis.
Counting a touch-point twice or confusing the number of terms with the number of zeroes.
"Zeroes never exceed the degree."
Practice questions
- 1A cubic polynomial can have at most ___ zeroes.
- 2The graph of y = p(x) cuts the x-axis at 6 different points. What is the least possible degree of p(x)?
- 3What are the zeroes of the cubic polynomial p(x) = (x − 6)(x − 3)(x + 2)?
- 4What are the zeroes of the cubic polynomial p(x) = (x − 5)(x + 5)(x + 4)?
- 5What are the zeroes of the cubic polynomial p(x) = (x − 3)(x − 1)(x + 1)?
- 6What are the zeroes of the cubic polynomial p(x) = (x + 6)(x + 2)(x + 5)?
- 7How many different zeroes does the cubic polynomial x³ − 9x have?
- 8How many different zeroes does the cubic polynomial x³ + 8x have?
Show answers (try them yourself first)
- 13
- 26
- 36, 3, −2
- 45, −5, −4
- 53, 1, −1
- 6−6, −2, −5
- 73
- 81