Problem 87 - Algebra A Combined Function

Question

Simplify each polynomial by combining like terms $$1.85 x^{2}-3.76 x+9.25 x^{2}+10.76-4.21 x$$

Step-by-Step Solution

Answer

The simplified polynomial is \(11.10x^2 - 7.97x + 10.76\).

Step 1: Identify Like Terms

Start by identifying terms of the polynomial that have the same variables raised to the same powers. In the given polynomial, we have two kinds of terms: the ones with \(x^2\) and the ones with \(x\). The like terms are \(1.85x^2\) and \(9.25x^2\), as well as \(-3.76x\) and \(-4.21x\). The constant \(10.76\) stands alone.

Step 2: Combine Like Terms for \(x^2\)

Add the coefficients of the like terms with \(x^2\). This means you add \(1.85\) and \(9.25\).\[ 1.85x^2 + 9.25x^2 = 11.10x^2 \]

Step 3: Combine Like Terms for \(x\)

Add the coefficients of the like terms with \(x\). In this case, you add \(-3.76\) and \(-4.21\).\[ -3.76x - 4.21x = -7.97x \]

Step 4: Write the Simplified Expression

Combine the results from Steps 2 and 3 along with the constant term \(10.76\). Thus, the simplified polynomial is:\[ 11.10x^2 - 7.97x + 10.76 \]

Key Concepts

Combining like terms

Simplifying polynomials often requires the process of combining like terms. This is one of the most fundamental skills in algebra and it involves grouping together terms that have the same variable raised to the same power. Think of like terms as ingredients in a recipe that can only be combined if they are the same type.

• For example, in the polynomial \(1.85x^2 - 3.76x + 9.25x^2 + 10.76 - 4.21x\), the like terms are \(1.85x^2\) and \(9.25x^2\), as both contain \(x^2\).
• Similarly, \(-3.76x\) and \(-4.21x\) are like terms because both contain \(x\).

Next, you simply add or subtract the coefficients (the numbers in front of the variables) of these like terms.

• For \(x^2\) terms, add \(1.85 + 9.25\) to get \(11.10x^2\).
• For \(x\) terms, add \(-3.76 + (-4.21)\) which equals \(-7.97x\). Remember when combining negative numbers you add their absolute values and keep the negative sign.

Polynomial expressions

Polynomial expressions are mathematical phrases that involve sums, differences, or zero or more variables raised to whole number exponents. These expressions can include terms with different powers, such as \(x\), \(x^2\), \(x^3\), and so forth.In our example, \(1.85x^2 - 3.76x + 9.25x^2 + 10.76 - 4.21x\), you can notice how there are terms related to \(x^2\) and \(x\), plus a constant term \(10.76\), which is just a number on its own without any variables.Polynomial expressions can be simplified by combining the like terms. This helps in reducing the complexity of the expression and makes it easier to solve or evaluate. Simplifying a polynomial also involves making sure that each term is ordered by their degree, usually starting with the highest power descending to the constant term.

Algebraic simplification

Algebraic simplification is about reducing expressions to their simplest form. This makes complex problems easier to interpret and solve. The process usually involves performing operations like combining like terms, distributing, and sometimes factoring.For polynomials, simplification means combining like terms and writing the expression in its simplest form. In the example provided, simplifying \(1.85x^2 - 3.76x + 9.25x^2 + 10.76 - 4.21x\) results in combining coefficients for like terms to produce a more straightforward expression \(11.10x^2 - 7.97x + 10.76\).To practice effective algebraic simplification:

• Start by identifying and combining all like terms.
• Make sure each step maintains equality, meaning do the same operation on both sides if it's an equation.
• Check your work by substituting back to ensure the simplified expression equals the original expression when solved for a specific variable.

Simplified expressions are not only easier to work with, but they also reduce the chances of errors in subsequent calculations.