When the machine prints zero

The idea

A real number k is a zero of a polynomial p(x) if p(k) = 0.

Zero on the display

Feed p(x) = x² − 3x − 4 any whole number from −5 to 5. Some inputs make it print exactly 0.

How many of those inputs will print 0?

• None
• Exactly one
• More than one

Hunt for 0

Feed numbers until the machine prints 0.

Zero of a polynomial

A real number k is a zero of the polynomial p(x) if p(k) = 0.

So −1 and 4 are zeroes of x² − 3x − 4, because p(−1) = 0 and p(4) = 0.

A zero is the number you put in, not the value that comes out. And a zero does not have to be 0: here the zeroes are −1 and 4.

Notes

A real number k is a zero of p(x) if p(k) = 0. Example: −1 and 4 are zeroes of x² − 3x − 4.

Check yourself

Is 3 a zero of p(x) = x² − 5x + 6?

x² − 9 has a negative zero. What is it?

Answer: -3

p(−3) = (−3)² − 9 = 9 − 9 = 0. So −3 is a zero.

For some polynomial, p(2) = 0. Which is true?

Which number is a zero of p(x) = x − 7?

• Yes, because p(3) = 9 − 15 + 6 = 0 — correct. Yes! p(3) = 0, so 3 is a zero.
• No, because p(3) = 6. Work it out: 9 − 15 + 6 = 0, not 6.
• No, because 3 is not 0. A zero need not be 0. What matters is that p(3) = 0, and it is.
• 2 is a zero of p(x) — correct. Yes! p(2) = 0 is exactly what “2 is a zero” means.
• The zero of p(x) is p(2). p(2) is the value that came out (here 0). The zero is the number that went in: 2.
• p(x) = 2 for every x. No. Only the value at x = 2 is known: it is 0.
• 7 — correct. Yes! p(7) = 7 − 7 = 0.
• 0. p(0) = 0 − 7 = −7, not 0.
• −7. p(−7) = −7 − 7 = −14, not 0.
• 1. p(1) = 1 − 7 = −6, not 0.