Polynomials – SS2 Further Mathematics Lesson Note (First Term)

A polynomial is an expression made from non-negative whole-number powers of a variable x, multiplied by numerical constants and added together.

Behavioural objectives

By the end of the lesson, students should be able to:

1. write the general form of a polynomial and name its coefficients, constant term and degree;
2. name a polynomial by its degree (constant, linear, quadratic, cubic or quartic);
3. identify expressions that are polynomials and expressions that are not;
4. state the condition for two polynomials to be equal;
5. add and subtract polynomials.

What is a polynomial?

A polynomial in x is an expression in which every power of x is a non-negative whole number, usually written p(x) or P(x).

The general polynomial has this form:

P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₂x² + a₁x + a₀

Here a_n, a_n-1, …, a₀ are numerical constants and a_n ≠ 0.

What are the parts of a polynomial?

• Variable: the letter x, which can take different values.
• Term: each part joined by + or -, such as 3x² or -2x.
• Coefficient: the constant that multiplies a power of x. In 3x² – 2x + 4, the coefficient of x² is 3 and of x is -2.
• Constant term: the term with no x, written a₀. In 3x² – 2x + 4 it is 4.
• Degree: the highest power of the variable, written n in the general form. In 3x² – 2x + 4 the degree is 2.
• Leading coefficient: the coefficient of the highest power, a_n. It cannot be zero.

Examples of polynomials

• (a) P₁(x) = 3x² – 2x + 4: degree 2.
• (b) P₂(x) = 3x² – 2x² + x – 1: simplifies to x² + x – 1, so the degree is 2.
• (c) P₃(x) = x + 1: degree 1.
• (d) P₄(x) = 2x³ + x – 3: degree 3.

What are the types of polynomial by degree?

Polynomials are named by their degree: constant, linear, quadratic, cubic and quartic.

Which expressions are not polynomials?

An expression is not a polynomial if x has a negative power, a fractional power or appears in a denominator.

• Negative power: (2x – 3)^-1, which is the same as 1/(2x – 3), is not a polynomial.
• Fractional power: (x² + 2x – 3)^1/2, the square root of x² + 2x – 3, is not a polynomial.
• Variable in the denominator: (3x² – 4x² + 2x – 1)/(2x + 3) is not a polynomial.

When are two polynomials equal?

Two polynomials are equal when the coefficients of every matching power of x are equal.

Let P(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀ and Q(x) = b_nxⁿ + b_n-1x^n-1 + … + b₁x + b₀. Then P(x) = Q(x) provided:

a_n = b_n, a_n-1 = b_n-1, …, a₂ = b₂, a₁ = b₁, a₀ = b₀

• Missing term: a power of x that is absent has coefficient 0.
• Example: if ax² + bx + 3 = 2x² – 5x + c, then a = 2, b = -5 and c = 3.

How do we add and subtract polynomials?

To add or subtract polynomials, add or subtract the coefficients of like powers of x.

P(x) + Q(x) = (a_n + b_n)xⁿ + (a_n-1 + b_n-1)x^n-1 + … + (a₁ + b₁)x + (a₀ + b₀)

P(x) – Q(x) = (a_n – b_n)xⁿ + (a_n-1 – b_n-1)x^n-1 + … + (a₁ – b₁)x + (a₀ – b₀)

What are the steps?

1. Write each polynomial in brackets, with the sign between them.
2. For subtraction, change the sign of every term in the second bracket.
3. Group like terms: the x³ terms, the x² terms, the x terms and the constants.
4. Add the coefficients in each group.
5. Write the answer from the highest power to the lowest.

• Multiplying by a constant: multiply every coefficient by that constant, so 3(2x² + x – 6) = 6x² + 3x – 18.

Worked examples

Given that P₁(x) = 7x³ – 4x² + 3x + 4, P₂(x) = 5x² + 6x + 1 and P₃(x) = 4x³ + 2x – 3, find:

• (a) P₁(x) + P₂(x)
• (b) P₁(x) + P₃(x)
• (c) P₁(x) – P₂(x)
• (d) P₃(x) – P₂(x)
• (e) P₁(x) + P₂(x) + P₃(x)

Solution

(a) P₁(x) + P₂(x)

= (7x³ – 4x² + 3x + 4) + (5x² + 6x + 1)
= 7x³ + (-4x² + 5x²) + (3x + 6x) + (4 + 1)
= 7x³ + x² + 9x + 5

(b) P₁(x) + P₃(x)

= (7x³ – 4x² + 3x + 4) + (4x³ + 2x – 3)
= (7x³ + 4x³) + (-4x²) + (3x + 2x) + (4 – 3)
= 11x³ – 4x² + 5x + 1

(c) P₁(x) – P₂(x)

= (7x³ – 4x² + 3x + 4) – (5x² + 6x + 1)
= 7x³ + (-4x² – 5x²) + (3x – 6x) + (4 – 1)
= 7x³ – 9x² – 3x + 3

(d) P₃(x) – P₂(x)

= (4x³ + 2x – 3) – (5x² + 6x + 1)
= 4x³ – 5x² + (2x – 6x) + (-3 – 1)
= 4x³ – 5x² – 4x – 4

(e) P₁(x) + P₂(x) + P₃(x)

= (7x³ – 4x² + 3x + 4) + (5x² + 6x + 1) + (4x³ + 2x – 3)
= (7x³ + 4x³) + (-4x² + 5x²) + (3x + 6x + 2x) + (4 + 1 – 3)
= 11x³ + x² + 11x + 2

Evaluation

1. State the degree, the constant term and the coefficient of x in 3x² – 2x + 4.
2. State the condition for two polynomials P(x) and Q(x) to be equal.
3. Using P₂(x) and P₃(x) from the worked examples, find P₂(x) + P₃(x).
4. Using P₁(x) and P₃(x) from the worked examples, find P₃(x) – P₁(x).
5. Give one reason why 1/(2x – 3) is not a polynomial.

Assignment

If p₁ = 3x³ + 2x² – x + 2, p₂ = 2x² + x – 6 and p₃ = x³ + 3x² + 2x – 4, find:

1. p₁ + p₂
2. p₃ – p₂
3. p₂ + p₃ – 3p₁
4. p₁ + p₂ + p₃