Adding, subtracting and multiplying polynomials

Warm-up

Three quick questions from earlier lessons.

Discover

A box made from a 20 cm by 30 cm sheet, with squares of side $x$ cut from the corners, has volume $x(20 - 2x)(30 - 2x)$ . Multiplied out, what is the highest power of $x$ ?

$x^3$
$x^2$

Key idea

Polynomials combine like numbers.

To add or subtract, combine like terms (same power of

x

). To multiply, multiply every term of one by every term of the other, then combine like terms. The degree of a product is the sum of the degrees.

Worked example

$(x + 3)(x^2 - 2x + 5)$ .

x^3 - 2x^2 + 5x

x times each term.

3x^2 - 6x + 15

3 times each term.

x^3 + x^2 - x + 15

Combine like terms.

Your turn, guided

Your turn: the coefficient of $x$ in $(x + 4)(x + 5)$ .

9

5x + 4x.

Practice ★

Expand $(x + 6)(x + 9)$ . What is the coefficient of $x$ ?
Expand $(x + 1)(x + 5)$ . What is the coefficient of $x$ ?
Expand $(x + 7)(x + 3)$ . What is the coefficient of $x$ ?

Watch out!

Practice ★★

What is the constant term of $(x + 9)(x^2 + 1x + 6)$ ?
What is the constant term of $(x + 7)(x^2 + 7x + 3)$ ?

A polynomial of degree 5 is multiplied by a polynomial of degree 5. What is the degree of the product?

Exit ticket

Three questions to check you have got it.

Expand $(x + 6)(x + 9)$ . What is the coefficient of $x$ ?
Expand $(x + 1)(x + 5)$ . What is the coefficient of $x$ ?
Expand $(x + 7)(x + 3)$ . What is the coefficient of $x$ ?

Summary card

Polynomial arithmetic

Add: combine like terms.
Multiply: every term by every term; degrees add.

Watch out: $x$ -terms add; constants multiply.

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