Sure! Let's address each question step-by-step.
### Q2. Coefficient of [tex]\( x^2 \)[/tex] and leading term in the polynomial [tex]\( 5-7 x^2+7 x^3+\sqrt{11} x^5 \)[/tex]:
1. Identify the coefficient of [tex]\( x^2 \)[/tex]:
- In the polynomial [tex]\( 5 - 7 x^2 + 7 x^3 + \sqrt{11} x^5 \)[/tex], we look at each term.
- The term involving [tex]\( x^2 \)[/tex] is [tex]\( -7 x^2 \)[/tex].
- The coefficient of [tex]\( x^2 \)[/tex] is therefore [tex]\( -7 \)[/tex].
2. Determine the leading term:
- The leading term of a polynomial is the term with the highest degree (the highest power of [tex]\( x \)[/tex]).
- In the polynomial [tex]\( 5 - 7 x^2 + 7 x^3 + \sqrt{11} x^5 \)[/tex], the term with the highest degree is [tex]\( \sqrt{11} x^5 \)[/tex].
- Therefore, the leading term is [tex]\( \sqrt{11} x^5 \)[/tex].
Hence, the coefficient of [tex]\( x^2 \)[/tex] is [tex]\( -7 \)[/tex] and the leading term is [tex]\( \sqrt{11} x^5 \)[/tex].
### Q3. Find the roots of the polynomial equation [tex]\( (x+3)(x+2)=0 \)[/tex]:
1. Setting the polynomial equal to zero:
- We start with the equation [tex]\( (x + 3)(x + 2) = 0 \)[/tex].
2. Using the Zero Product Property:
- The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero.
- So, we can set each factor equal to zero: [tex]\( x + 3 = 0 \)[/tex] and [tex]\( x + 2 = 0 \)[/tex].
3. Solving for [tex]\( x \)[/tex] in each equation:
- For [tex]\( x + 3 = 0 \)[/tex]:
[tex]\[
x + 3 = 0 \implies x = -3
\][/tex]
- For [tex]\( x + 2 = 0 \)[/tex]:
[tex]\[
x + 2 = 0 \implies x = -2
\][/tex]
Therefore, the roots of the polynomial equation [tex]\( (x + 3)(x + 2) = 0 \)[/tex] are [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex].
In summary:
- The coefficient of [tex]\( x^2 \)[/tex] in the polynomial [tex]\( 5 - 7 x^2 + 7 x^3 + \sqrt{11} x^5 \)[/tex] is [tex]\( -7 \)[/tex].
- The leading term in the polynomial is [tex]\( \sqrt{11} x^5 \)[/tex].
- The roots of the polynomial equation [tex]\( (x + 3)(x + 2) = 0 \)[/tex] are [tex]\( x = -3 \)[/tex] and [tex]\( x = -2 \)[/tex].
