L09 — Introduction to Polynomials
Key Terms
- Polynomial — a sum of terms of the form axn where n is a non-negative integer and a is a real number
- Degree — the highest power of x with a non-zero coefficient
- Leading coefficient — the coefficient of the highest-degree term
- Standard form — terms written in descending order of degree: anxn + an−1xn−1 + … + a0
- Evaluate P(a) — substitute x = a into the polynomial and calculate the result
- End behaviour — how the graph of P(x) behaves as x → +∞ or x → −∞
What is a Polynomial?
A polynomial in x is an expression of the form:
anxn + an−1xn−1 + … + a1x + a0
where n is a non-negative integer and the coefficients an, …, a0 are real numbers with an ≠ 0.
Degree and Classification
| Name | Degree | General Form | Example |
|---|---|---|---|
| Constant | 0 | a | 7 |
| Linear | 1 | ax + b | 3x − 2 |
| Quadratic | 2 | ax² + bx + c | x² + 5x − 3 |
| Cubic | 3 | ax³ + bx² + cx + d | 2x³ − x + 4 |
| Quartic | 4 | ax4 + … | x4 − 3x² + 1 |
- The degree is the highest power of x with a non-zero coefficient.
- The leading coefficient is the coefficient of the highest-degree term.
- Standard form means terms are written in descending order of degree.
End Behaviour
| Degree | Leading Coeff | As x → −∞ | As x → +∞ |
|---|---|---|---|
| Odd | Positive (+) | y → −∞ | y → +∞ |
| Odd | Negative (−) | y → +∞ | y → −∞ |
| Even | Positive (+) | y → +∞ | y → +∞ |
| Even | Negative (−) | y → −∞ | y → −∞ |
The end behaviour is determined entirely by the degree and sign of the leading coefficient.
Evaluating a Polynomial
P(−2) = (−2)³ − 2(−2) + 3 = −8 + 4 + 3 = −1
What Makes Something a Polynomial?
A polynomial is a sum of terms of the form axn where n must be a non-negative integer (0, 1, 2, 3, …). The key restrictions are:
- No negative powers of x (so 1/x = x−1 is not allowed)
- No fractional powers of x (so √x = x½ is not allowed)
- Coefficients can be any real number (including fractions and negative numbers)
Classify each expression as a polynomial or not a polynomial. If it is a polynomial, state its degree and leading coefficient.
(a) 3x&sup4; − x² + 7 (b) x² + √x + 1 (c) −5x³ + 2x − 4 (d) 2/x + x
Solution:
(a) Polynomial. Degree 4, leading coefficient 3.
(b) Not a polynomial — √x = x½ has a fractional exponent.
(c) Polynomial. Degree 3, leading coefficient −5.
(d) Not a polynomial — 2/x = 2x−1 has a negative exponent.
Standard Form and Evaluating Polynomials
Standard form lists terms in descending order of degree. For example, 3 + 2x − x² in standard form is −x² + 2x + 3.
To evaluate P(a), substitute every x with the value a and calculate carefully. Use brackets around negative values.
For P(x) = 2x³ − x + 4, find P(0), P(2), and P(−1).
Solution:
P(0) = 2(0)³ − (0) + 4 = 0 − 0 + 4 = 4
P(2) = 2(2)³ − (2) + 4 = 16 − 2 + 4 = 18
P(−1) = 2(−1)³ − (−1) + 4 = −2 + 1 + 4 = 3
Polynomial Arithmetic
To add or subtract polynomials, collect like terms (terms with the same degree). To multiply, use the distributive law and remember that xm × xn = xm+n.
(a) (4x³ − 2x + 1) + (x³ + 3x² + x − 5)
(b) (3x² + 5x − 2) − (x² − 2x + 4)
Solution:
(a) Collect like terms: (4+1)x³ + 3x² + (−2+1)x + (1−5) = 5x³ + 3x² − x − 4
(b) Distribute the minus: (3−1)x² + (5+2)x + (−2−4) = 2x² + 7x − 6
End Behaviour
For large values of |x|, a polynomial is dominated by its highest-degree term. So the end behaviour depends only on the degree and the sign of the leading coefficient:
- Odd degree, positive leading coeff: falls left (→ −∞), rises right (→ +∞)
- Odd degree, negative leading coeff: rises left, falls right
- Even degree, positive leading coeff: rises at both ends (→ +∞ both sides)
- Even degree, negative leading coeff: falls at both ends (→ −∞ both sides)
State the end behaviour of y = −3x4 + 5x² − x + 1.
Solution:
Degree = 4 (even). Leading coefficient = −3 (negative).
⇒ As x → −∞, y → −∞ and as x → +∞, y → −∞.
The graph falls on both ends (like an upside-down U overall).
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Degree and leading coefficient. Fluency
For each polynomial, state the degree and the leading coefficient.
- (a) 3x² − 2x + 5
- (b) x5 − 4x³ + 2
- (c) −2x³ + x − 7
- (d) 6x4 − 3x² + x
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Standard form. Fluency
Rewrite each polynomial in standard form (descending powers of x). State the degree.
- (a) 3 + 2x − x²
- (b) 4x³ − 2 + x − 5x²
- (c) x + 3x4 − x²
- (d) 5 − 3x + 7x² − x³
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Evaluate the polynomial. Fluency
For P(x) = 2x³ − 3x² + x − 4, evaluate:
- (a) P(0)
- (b) P(1)
- (c) P(−1)
- (d) P(2)
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Add and subtract polynomials. Fluency
Expand the brackets and collect like terms.
- (a) (3x² − 2x + 1) + (x² + 4x − 3)
- (b) (5x³ − 2x + 4) − (2x³ + x² − 3x + 1)
- (c) (2x4 − x² + 3x) + (x³ + 4x² − x − 5)
- (d) (x³ − 4x² + 2x − 1) − (x³ + 2x − 5)
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Multiply polynomials. Understanding
Expand and simplify.
- (a) x(x² − 3x + 2)
- (b) (x + 2)(x² − x + 3)
- (c) (x − 1)(x + 1)(x − 2)
- (d) (2x + 1)(x² − 2x + 3)
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End behaviour. Understanding
For each polynomial, use the degree and leading coefficient to describe the end behaviour as x → −∞ and as x → +∞.
- (a) y = x³ − 2x + 1
- (b) y = −x4 + 3x²
- (c) y = 2x5 − x³
- (d) y = −3x² + 5x − 1
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Polynomial from factored form. Understanding
For P(x) = (x − 1)(x + 2)(x − 3):
- (a) State the x-intercepts of the graph.
- (b) Expand to standard form.
- (c) Find the y-intercept (the value of P(0)).
- (d) Describe the end behaviour.
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Does the point lie on the polynomial? Understanding
P(x) = x³ − 2x² − x + 2. Determine whether each point lies on the graph of P.
- (a) (0, 2)
- (b) (1, 0)
- (c) (2, 1)
- (d) (−1, 0)
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Find the unknown coefficient. Problem Solving
P(x) = kx³ − 3x² + 2x − 1. Given that P(2) = 5, find the value of k.
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Construct a polynomial from zeros. Problem Solving
A cubic polynomial has the form P(x) = k(x − a)(x − b)(x − c). It has x-intercepts at x = −2, x = 0, and x = 3, and passes through the point (1, 12).
- (a) Write the unscaled factored form P(x) = k ⋅ x(x + 2)(x − 3).
- (b) Use the point (1, 12) to find k.
- (c) Write the full equation of P(x) in standard form.