Question

Explain the mistake and work the problem correctly. $$\begin{array}{l} \left(-9 t^{2}+4 t+11\right)-\left(5 t^{3}+9 t^{2}+6\right) \\ \quad=\left(-9 t^{2}+4 t+11\right)+\left(-5 t^{3}-9 t^{2}-6\right) \\ \quad=-5 t^{3}+4 t+5 \end{array}$$

Short answer

Question: Subtract the second polynomial expression from the first one. Expression 1: $$-9t^2 + 4t + 11$$ Expression 2: $$5t^3 + 9t^2 + 6$$ Provide the simplified result of the subtraction. Answer: $$-5t^3 - 18t^2 + 4t + 5$$

Step-by-step solution

Identify the given expressions

We have two polynomial expressions: Expression 1: $$-9t^2 + 4t + 11$$ Expression 2: $$5t^3 + 9t^2 + 6$$ The task is to subtract Expression 2 from Expression 1.

Write the task in mathematical notation

To represent the task mathematically, write it as: $$(-9t^2 + 4t + 11) - (5t^3 + 9t^2 + 6)$$

Distribute the negative sign to the second expression

To distribute the negative sign to the second expression, change the signs of each term within the expression: $$(-9t^2 + 4t + 11) + (-5t^3 - 9t^2 - 6)$$

Combine like terms

Now, combine the like terms in the expression: $$-5t^3 + (-9t^2 - 9t^2) + (4t) + (11 - 6)$$

Simplify the expression

Finally, simplify the expression by combining the terms: $$-5t^3 - 18t^2 + 4t + 5$$ So, the correct result is: $$-5t^3 - 18t^2 + 4t + 5$$

Key concepts

Combining Like Terms

When we work with polynomial expressions, a fundamental step in simplifying is to combine like terms. Like terms are terms in an algebraic expression that have the same variables raised to the same powers, even if their coefficients are different. For example, in the expression \(-5t^3 - 18t^2 + 4t + 5\), the terms \(-5t^3\) and \(-18t^2\) are not like terms because they involve the variable \(t\) raised to different powers.

To combine like terms correctly, we simply add or subtract their coefficients. As demonstrated in the exercise solution, combining the like terms \(-9t^2\) and \(-9t^2\) results in \(-18t^2\), while the constants 11 and -6 combine to form 5. Always ensure you're only combining terms that match in variables and exponents, otherwise, it could lead to incorrect results.

Distributive Property

The distributive property is a critical algebraic rule that helps us simplify expressions and especially comes into play during polynomial subtraction. The property states that for any real numbers \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. This property also works when distributing a negative sign across a polynomial, as was required in our exercise.

When the problem asked us to subtract one polynomial from another, effectively, we had to distribute a negative one across the second polynomial before subtraction, which flips the signs of all its terms. For instance, \(-(5t^3 + 9t^2 + 6)\) becomes \(-5t^3 - 9t^2 - 6\). Remembering to apply the distributive property correctly can make the difference between a correct and incorrect solution.

Polynomial Expressions

Polynomial expressions are algebraic expressions that include variables and constants. The variables in polynomials can have whole number exponents (or be to the power of zero, implying a constant). An important note about polynomials is that the expressions should have terms with non-negative integer exponents; otherwise, the expression may not qualify as a polynomial.

Looking at our original exercise, the expressions \(-9t^2 + 4t + 11\) and \(5t^3 + 9t^2 + 6\) are both examples of polynomial expressions. Each expression is made up of a sum of terms, where each term is a product of a coefficient and a variable raised to a power. Polynomials can be classified by their degree, which is the highest power of the variable that appears in the expression.

Simplifying Expressions

Simplifying expressions, like the one in our exercise, involves several steps to reduce an algebraic expression to its simplest form. This typically includes distributing any necessary negative or positive signs, combining like terms, and applying any other relevant algebraic properties.

In the context of our example, to simplify \(-9t^2 + 4t + 11 - (5t^3 + 9t^2 + 6)\), we started by distributing the negative sign across the second polynomial, then combined like terms, and arrived at the simplified polynomial \(-5t^3 - 18t^2 + 4t + 5\). Simplifying expressions helps us understand the structure of the expression and can make further algebraic operations, like factoring or graphing, much easier.