Undo the b, undo the a

The idea

The zero of the linear polynomial ax + b is –b/a, that is, –(Constant term)/(Coefficient of x).

Zero of 2x + 3

To find the zero of 2x + 3, ask: which x makes it equal 0?

The zero of 2x + 3 is…

• −3/2
• 3/2
• −2/3
• 3

Hit the target

Choose a and b for the polynomial ax + b. The working shows its zero.

The zero is −b/a

The zero of the linear polynomial ax + b is −b/a that is, −(constant term)/(coefficient of x).

Why? If k is a zero, then p(k) = ak + b = 0. So ak = −b, and k = −b/a.

Example: for 2x + 3, a = 2 and b = 3, so the zero is −3/2. Check: 2 × (−3/2) + 3 = −3 + 3 = 0.

The zero of a linear polynomial is built from its coefficients. Is the same true for other polynomials? That is the question for this chapter.

Notes

The zero of the linear polynomial ax + b is −b/a that is, −(constant term)/(coefficient of x).

Check yourself

Find the zero of 5x − 10.

Answer: 2

−b/a = −(−10)/5 = 10/5 = 2. Check: 5 × 2 − 10 = 0.

Find the zero of 3x + 12.

Answer: -4

−b/a = −12/3 = −4. Check: 3 × (−4) + 12 = 0.

Which linear polynomial has the zero 5?

Why must a ≠ 0 in ax + b?

• x − 5 — correct. Yes! a = 1, b = −5, so −b/a = 5.
• x + 5. Its zero is −5.
• 5x. Its zero is 0.
• 5x + 1. Its zero is −1/5.
• If a = 0 there is no x left and we cannot divide by a — correct. Yes! With a = 0, ax + b is just b, and −b/a makes no sense.
• Because a = 0 makes the zero equal 0. It makes −b/a meaningless: we cannot divide by 0.
• Because b must be 0. b can be any real number. Only a must not be 0.