Question

Which of the following is/are true about the expression $4x^3+2x^{\frac{5}{2}}(x^{\frac{3}{2}}+3x^{\frac{1}{2}})+7?$

• A. Not a polynomial
• B. A polynomial
• C. Degree is $4$
• D. Degree is not defined

Answer 1

Answer: (B), (C)

Degree is $4$

Let's simplify the given expression,
$4x^3+2x^{\frac{5}{2}}(x^{\frac{3}{2}}+3x^{\frac{1}{2}})+7$

Applying distributive property of multiplication: $a\times (b+c)=a\times b+a\times c$

$=4x^3+2x^{\frac{5}{2}}\cdot x^{\frac{3}{2}}+2x^{\frac{5}{2}}\cdot 3x^{\frac{1}{2}}+7$

$=4x^3+2x^{\frac{5}{2}}\cdot x^{\frac{3}{2}}+2\cdot 3\cdot x^{\frac{5}{2}}\cdot x^{\frac{1}{2}}+7$

$=4x^3+2x^{(\frac{5}{2}+\frac{3}{2})}+2\cdot 3\cdot x^{(\frac{5}{2}+\frac{1}{2})}+7$

$[\because a^m\cdot a^n=a^{(m+n)}]$

$=4x^3+2x^{(\frac{5+3}{2})}+6x^{(\frac{5+1}{2})}+7$

$=4x^3+2x^{(\frac{8}{2})}+6x^{(\frac{6}{2})}+7$

$=4x^3+2x^{(\frac{4\times 2}{2})}+6x^{(\frac{3\times 2}{2})}+7$

$=4x^3+2x^4+6x^3+7$

Combining the like terms

$=\underset{–}{4x^3+6x^3}+2x^4+7$

$=10x^3+2x^4+7$

Arranging the terms in descending order of their powers

$=2x^4+10x^3+7$

As a result, given expression is simplified to $\underset{–}{2x^4+10x^3+7}.$

In the simplified expression $\underset{–}{2x^4+10x^3+7},$ the powers of variable belongs to set of whole numbers. Therefore, given expression is a $\underset{–}{\text{polynomial}}$.

Now, as we know, the largest power of the variable is the degree of the polynomial. Here, the largest power of the variable $\underset{–}{x}$ is $\underset{–}{4}.$ Therefore, degree of the given polynomial is $\underset{–}{4}.$