The constant **polynomial** whose coefficients are all equal to 0. The corresponding **polynomial** function is the constant function with value 0, also called the **zero** map. The **zero polynomial** is the additive identity of the additive group of **polynomials**.

Consider the polynomial p(x) = 5x^{3}– 2x^{2} + 3x – 2.

If we replace x by 1 everywhere in p(x), we get

p(1) = 5 × (1)^{3} – 2 × (1)^{2} + 3 × (1) – 2

= 5 – 2 + 3 –2

= 4

So, we say that the value of p(x) at x = 1 is 4.

Similarly, p(0) = 5(0)^{3} – 2(0)^{2} + 3(0) –2 = –2

**Example: Find the value of each of the following polynomials at the indicated value of variables:**

**(i) p(x) = 5x ^{2} – 3x + 7 at x = 1**

**(ii) q(y) = 3y ^{3} – 4y + √11 at y = 2**

**(iii) p(t) = 4t ^{4} + 5t^{3} – t^{2} + 6 at t = a**

Solution:

(i) p(x) = ** **5x^{2} – 3x + 7

The value of the polynomial p(x) at x = 1 is given by

p(1) = 5(1)^{2} – 3(1) + 7 = 5 – 3 + 7 = 9

(ii) q(y) = 3y^{3} – 4y + √11

The value of the polynomial q(y) at x = 2 is given by

q(2) = 3(2)^{3} – 4(2) + √11 = 3×8 – 8 + √11 = 24 – 8 + √11 = 16 + √11

(iii) p(t) = 4t^{4} + 5t^{3} – t^{2} + 6 at t = a

The value of the polynomial p(t) at t = a is given by

p(a) = 4a^{4} + 5a^{3} – a^{2} + 6

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