Polynomials Test

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Polynomials Test - 38

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Weekly Quiz Competition

Question 1
1 / -0

What is the degree of the polynomial $$p(x) = 5x^3 - 8x^2 + 4x?$$

A
$$3$$

B
$$2$$

C
$$1$$

D
$$0$$

Solution

The polynomial $$ax^3+ bx^2+cx+d=0$$ is in standard form.
On comparing $$5x^3-8x^2+4x$$ with the standard form, the power of the first term is $$3$$, the power of the second term is $$2$$ and the power of the third term is $$1$$. Since the polynomial has the largest exponent, that is $$3$$ which is the degree of the polynomial.

Hence, the degree of the polynomial is $$3$$
Question 2
1 / -0

Zero polynomial can be written as ________.

A
$$p(x) = x$$

B
$$p(x) = 1$$

C
$$p(x) = 0$$

D
$$p(x) = x^2$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.

Therefore, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.

Hence, zero polynomial can be written as $$p(x) = 0$$.
Question 3
1 / -0

What is the degree of the polynomial $$p(x) = 8x^8 + 9x^9 + 10x^0$$?

A
$$8$$

B
$$9$$

C
$$10$$

D
$$0$$

Solution

After writing the $$9x^9+8x^8+10x^0$$ we can see that the power of the first term is $$9$$, the power of the second term is $$8$$ and the power of the third term is $$0$$. Since the polynomial has the largest exponent, that is $$9$$ which is the degree of the polynomial.

Hence, the degree of the polynomial is $$9$$.
Question 4
1 / -0

The constant polynomial whose coefficients are all equal to $$0$$ is called ________ polynomial.

A
Zero

B
Linear

C
Quadratic

D
Cubic

Solution

For example:-
Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.

Therefore, zero polynomial can be written as $$p(x) = 0$$.

Hence, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.
Question 5
1 / -0

A polynomial whose coefficients are all equal to _______ is called zero polynomial.

A
$$1$$

B
$$2$$

C
$$3$$

D
$$0$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.

Therefore, zero polynomial can be written as $$p(x) = 0$$.

Hence, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.
Question 6
1 / -0

Which of the following polynomials, has $$p(6) = 36$$ value?

A
$$p(x) = x$$

B
$$p(x) = x^2$$

C
$$p(x) = x^3$$

D
$$p(x) = x^0$$

Solution

Let $$p(x)=x^2$$ and substitute $$x=6$$ as shown below:

$$p(6)=(6)^{ 2 }=36$$

Hence, $$p(x)=x^2$$.
Question 7
1 / -0

For $$p(x) = 3x^2 - 5x, p(6) = $$

A
$$36$$

B
$$72$$

C
$$78$$

D
$$77$$

Solution

The polynomial given to us is $$p(x)=3x^2-5x$$ after substituting the value of $$x=6$$ in the polynomial:
$$p(6)=3(6)^2-(5\times 6)=(3\times 36)-30=108-30=78$$
Hence, $$p(6)=78$$.
Question 8
1 / -0

Which of the following does NOT represent a zero polynomial?

A
$$p(x) = 0$$

B
$$p(x) = 0.x^0$$

C
$$p(x) = x^0$$

D
$$p(x) = x^0 - 1$$

Solution

Consider the polynomial, $$p(x) = ax^2 + bx +c$$ , if $$a=b=c= 0$$ then the expression becomes zero polynomial.
Therefore, the constant polynomial whose coefficients are all equal to $$0$$ is called a zero polynomial.

Zero polynomial can be written as $$p(x) = 0$$.

Now, the polynomial $$p(x)=x^0$$ or $$p(x)=1$$ is a constant polynomial but the coefficient is not equal to $$0$$ and therefore it is not of the form $$p(x) = 0$$.

Hence, the polynomial $$p(x)=x^0$$ is not a zero polynomial
Question 9
1 / -0

The zero polynomial is the _________ identity of the additive group of polynomials.

A
multiplicative

B
additive

C
multiplicative inverse

D
additive inverse

Solution

Zero polynomial has all coefficients $$=0$$
If $$p(x)$$ is a zero polynomial then $$p(x)=0$$
It is the additive identity of additive group of polynomials.

Hence, $$Op-B$$ is correct.
Question 10
1 / -0

A real number $$\alpha$$ is said to be a zero of the polynomial $$p(x)$$, if ___________.

A
$$p(\alpha) = 0$$

B
$$p(\alpha) = 1$$

C
$$p(\alpha) = x$$

D
$$p(\alpha) = \alpha$$

Solution

Any real number $$\alpha $$ is said to be a zero of polynomial $$p\left( x \right) $$ if $$p\left( \alpha \right) =0.$$
Zero can be seen as the value of $$x$$ for which the polynomial takes the value as zero.
So, if $$x^2-2x+1$$ is a polynomial, we know at $$x=1.$$
In this case, the value of $$x^2-2x+1=0,$$ hence $$x=1$$ can be said as zero of $$x^2-2x+1$$
Hence, the answer is $$p\left( \alpha \right) =0.$$

That is, option $$A$$ is correct.

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Polynomials Test - 38

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Polynomials Test - 38

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Question 1

$$3$$

$$2$$

$$1$$

$$0$$

Solution

Question 2

$$p(x) = x$$

$$p(x) = 1$$

$$p(x) = 0$$

$$p(x) = x^2$$

Solution

Question 3

$$8$$

$$9$$

$$10$$

$$0$$

Solution

Question 4

Zero

Linear

Quadratic

Cubic

Solution

Question 5

$$1$$

$$2$$

$$3$$

$$0$$

Solution

Question 6

$$p(x) = x$$

$$p(x) = x^2$$

$$p(x) = x^3$$

$$p(x) = x^0$$

Solution

Question 7

$$36$$

$$72$$

$$78$$

$$77$$

Solution

Question 8

$$p(x) = 0$$

$$p(x) = 0.x^0$$

$$p(x) = x^0$$

$$p(x) = x^0 - 1$$

Solution

Question 9

multiplicative

additive

multiplicative inverse

additive inverse

Solution

Question 10

$$p(\alpha) = 0$$

$$p(\alpha) = 1$$

$$p(\alpha) = x$$

$$p(\alpha) = \alpha$$

Solution

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