Question

select all the correct answers
which statements are true about the result of simplifying this polynomial?
t³(8 + 9t) − (t² + 4)(t² − 3t)
the simplified expression has a constant term.
the simplified expression is a polynomial.
the simplified expression has four terms.
the simplified expression is a trinomial.
the simplified expression is cubic.

Explanation:

Step1: Expand the polynomial

First, expand \( t^{3}(8 + 9t) \):

$$ t^{3}(8 + 9t)=8t^{3}+9t^{4} $$

Then, expand \( (t^{2}+4)(t^{2}-3t) \) using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

Step2: Subtract the two expanded polynomials

Now, subtract the second expanded polynomial from the first:

$$
LATEXBLOCK1
$$

Step3: Analyze the simplified polynomial

• Constant term: A constant term is a term without a variable. In \( 8t^{4}+11t^{3}-4t^{2}+12t \), there is no constant term (the last term is \( 12t \), which has a variable \( t \)). So the statement "The simplified expression has a constant term" is false.
• Is it a polynomial?: A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non - negative integer exponents of variables. The simplified expression \( 8t^{4}+11t^{3}-4t^{2}+12t \) meets this definition, so the statement "The simplified expression is a polynomial" is true.
• Number of terms: The simplified expression \( 8t^{4}+11t^{3}-4t^{2}+12t \) has four terms (\( 8t^{4} \), \( 11t^{3} \), \( - 4t^{2} \), \( 12t \)), so the statement "The simplified expression has four terms" is true.
• Trinomial?: A trinomial has three terms. Since our simplified expression has four terms, the statement "The simplified expression is a trinomial" is false.
• Degree of the polynomial: The degree of a polynomial is the highest power of the variable in the polynomial. The highest power of \( t \) in \( 8t^{4}+11t^{3}-4t^{2}+12t \) is 4, so it is a quartic (degree 4) polynomial, not cubic. So the statement "The simplified expression is cubic" is false.

Answer:

The simplified expression is a polynomial, The simplified expression has four terms (i.e., the options: "The simplified expression is a polynomial", "The simplified expression has four terms")