Question

8 Prove that if degree of f(x).g(x) is 1, then one of f(x).g(x) is a constant and the other is a linear polynomial
.
O Solve the systemy
X + v + v= b; x+z+V=CY+z+v= d.​

Answer 1

To prove:

The degree of $\mathsf{f(x).g(x)}$ is $\mathsf{1}$, only when one of $\mathsf{f(x)}$, $\mathsf{g(x)}$ is a constant and the other is a linear polynomial

Step-by-step proof:

Let us prove the given statement by some examples.

Let, $\mathsf{f(x)=ax+b}$ (a linear polynomial) and $\mathsf{g(x)=c}$ (a constant).

• $\mathsf{\therefore f(x).g(x)=(ax+b).c}$

• $\mathsf{\Rightarrow f(x).g(x)=acx+bc}$

• This is of degree $\mathsf{1}$ because the highest power of the variable involved in the polynomial is $\mathsf{1}$.

This proves the statement by an example.

Let us go against the statement with different assumptions.

Let, $\mathsf{f(x)=ax+b}$ and $\mathsf{g(x)=cx+d}$ (both linear polynomials).

• $\mathsf{\therefore f(x).g(x)=(ax+b).(cx+d)}$

• $\mathsf{\Rightarrow f(x).g(x)=acx^{2}+(ad+bc)x+bd}$

• This is of degree $\mathsf{2}$ because the highest power of the variable involved in the polynomial is $\mathsf{2}$.

From this, we can conclude that whenever we take two polynomials with degree $\mathsf{>0}$, then the degree of the polynomial (found by multiplying the two polynomials mentioned) is the sum of the degrees of the two polynomials.

Conclusion:

The degree of $\mathsf{f(x).g(x)}$ is $\mathsf{1}$, only when one of $\mathsf{f(x)}$, $\mathsf{g(x)}$ is a constant and the other is a linear polynomial. (Proved)

Answer 2

Answer:

Explanation:

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asd