Feed the machine

The idea

The value obtained by replacing x by a real number k in a polynomial p(x) is the value of p(x) at x = k, denoted by p(k).

A machine named p

p(x) = x² − 3x − 4 is like a machine. Put in a number, and a number comes out. Put in 2.

What comes out?

• −6
• 6
• −10

Feed it

Tap numbers to feed the machine. It puts your number in place of every x.

The value p(k)

If p(x) is a polynomial in x and k is any real number, the number you get by replacing x by k is called the value of p(x) at x = k. It is written p(k).

For p(x) = x² − 3x − 4: p(2) = 2² − 3 × 2 − 4 = −6, and p(0) = −4.

Tip: put a negative number in brackets, like (−1)², so its sign is not lost.

Notes

The value of p(x) at x = k is what you get by replacing x by k. It is written p(k).

Check yourself

p(x) = x² − 3x − 4. Find p(3).

Answer: -4

p(3) = 3² − 3 × 3 − 4 = 9 − 9 − 4 = −4.

p(x) = x² − 3x − 4. Find p(−2).

Answer: 6

p(−2) = (−2)² − 3 × (−2) − 4 = 4 + 6 − 4 = 6.

What does p(5) mean?

For p(x) = 4x³ − x² + 9, find p(0).

Answer: 9

p(0) = 4 × 0 − 0 + 9 = 9. When x = 0, only the number term is left.

• p times 5. p(5) is not a multiplication. The bracket holds the number put in.
• The value of the polynomial when x is replaced by 5 — correct. Yes! Every x becomes 5.
• The 5th term of p(x). No. It means: replace x by 5 everywhere, then work out the number.