Problem 70 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. \(\left(3 x^{2}-5\right)\left(-x^{2}+1\right)\)

Step-by-Step Solution

Answer

The result of the multiplication in standard form is: \[-3x^{4} + 8x^{2} - 5\]

Step 1: Distribute the Terms

Distribute each term in the first binomial \(\left(3 x^{2}-5\right)\) with each term in the second binomial \(\left(-x^{2}+1\right)\). Doing this yields: \[3x^{2}*(-x^{2}) + 3x^{2}*1 -5*(-x^{2}) -5*1\]

Step 2: Simplify

Simplify the multiplication from the previous step to get: \[-3x^{4} + 3x^{2} + 5x^{2} - 5\]

Step 3: Collect Like Terms

Next, collect the like terms, specifically the x squared terms, to get the standard form of the polynomial: \[-3x^{4} + 8x^{2} - 5\]

Key Concepts

Standard Form

In mathematics, expressing polynomials in standard form is crucial for clear presentation and ease of further computation. To write a polynomial in standard form, you should arrange the terms in descending order of their exponents. This means starting with the term with the highest degree and working your way down to the constant term, which has an implied zero exponent.

For example, the polynomial \[-3x^{4} + 8x^{2} - 5\] is correctly placed in standard form as the terms are ordered by the descending powers of \(x\). The highest power term is \(-3x^4\), followed by \(8x^2\), and finally, the constant term \(-5\).

Ensuring your polynomial is in standard form allows not just better organization, but also facilitates polynomial addition, subtraction, and comparison.

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions and perform operations like the one in the example given. It states that multiplying a sum by a number gives the same result as multiplying each addend individually and then adding the products. The property can be represented as: \[a(b + c) = ab + ac\].

When dealing with polynomials, the distributive property is applied to multiply two binomials. In the exercise provided, each term in the first binomial \((3x^2 - 5)\) is distributed over each term in the second binomial \((-x^2 + 1)\):

• \(3x^2\) is multiplied by \(-x^2\) and \(1\)
• \(-5\) is multiplied by \(-x^2\) and \(1\)

This operation expands the expression, preparing it for further simplification. Proper application of the distributive property is key to working with binomials and polynomials, paving the way for easier simplification and combination of like terms.

Combining Like Terms

Combining like terms is an essential step in simplifying polynomial expressions. Like terms are those that have the same variable raised to the same power. To combine them, you simply add or subtract the coefficients. The original exercise results in the expanded expression: \[-3x^4 + 3x^2 + 5x^2 - 5\].

Here, \(3x^2\) and \(5x^2\) are like terms because they both involve \(x^2\). To combine them, add their coefficients:

• \(3x^2 + 5x^2 = 8x^2\)

After combining like terms, the expression simplifies to: \[-3x^4 + 8x^2 - 5\].

This step is crucial for obtaining the polynomial in its simplest form, which is often required before solving, graphing, or integrating the polynomial. It also makes the expression clear and concise for interpretation.