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Understanding how to combine like terms in polynomials is an essential skill. When you have a polynomial expression, it often contains terms with similar variables raised to the same power. For example, in the polynomial expression \[ 8x^3 + 3x^3 + 2x^2 - 5x + 7x + 12 \] terms like \(8x^3\) and \(3x^3\) are considered 'like terms' because they both involve \(x\) raised to the third power.To simplify the expression, you sum the coefficients of these like terms. It's like adding apples to apples; you cannot add apples to oranges.- **Step 1:** Identify terms with the same variable and power.- **Step 2:** Add their coefficients together.For instance, combining \(8x^3\) and \(3x^3\) yields \(11x^3\), while \(-5x\) and \(7x\) combine to give \(2x\). Once all the like terms are combined, the expression becomes simplified.

The distributive property is a useful tool in simplifying expressions that involve addition or subtraction of polynomials, especially when a term is outside the parenthesis. It says that you should 'distribute' multiplication over addition or subtraction.Consider an expression such as:\[ (a + b) \ m = a \ m + b \ m \]The idea is to multiply every term inside the parenthesis by the term outside. For the expression \[- (3x^3 + 2x^2 + 7x + 12)\] you multiply each term by -1 to effectively subtract each part:- \(-1 \ 3x^3 = -3x^3\)- \(-1 \ 2x^2 = -2x^2\)- \(-1 \ 7x = -7x\)- \(-1 \ 12 = -12\)Applying the distributive property turns the initial expression into manageable pieces for further simplification.

Adding and subtracting polynomials is straightforward once you grasp the concept of combining like terms. Let's walk through the process using part (a) and part (b) from the exercise. When adding polynomials, as in \[(8x^3 - 5x) + (3x^3 + 2x^2 + 7x + 12)\] you simply add coefficients of like terms: - Add \(8x^3\) and \(3x^3\) to get \(11x^3\)- The lone \(2x^2\) stands as it is - Combine \(-5x\) with \(7x\) to get \(2x\)- The constant \(12\) remains unchangedSubtraction works similarly but requires careful attention to minus signs.For part (b):\[(8x^3 - 5x) - (3x^3 + 2x^2 + 7x + 12)\] you first distribute the negative sign across the second polynomial and follow through with similar steps:- From \(8x^3\) minus \(3x^3\), you get \(5x^3\)- \(-2x^2\) is from \(-2x^2\)- Combine \(-5x\) and \(-7x\) to get \(-12x\)- The constant \(-12\) comes from subtracting 12.These operations make handling polynomials systematic and predictable.

Multiplying polynomials involves expanding each term in one polynomial by each term in the other polynomial using the distributive property. Let's tackle this complex expression:\[(2x^2 - 6x + 11) (-8x^2 - 7x + 9)\]1. **Multiply First Terms:** - \((2x^2) (-8x^2) = -16x^4\) - \((2x^2) (-7x) = -14x^3\) - \((2x^2) (9) = 18x^2\)2. **Multiply Second Terms: (Distribute -6x):** - \((-6x) (-8x^2) = 48x^3\) - \((-6x) (-7x) = 42x^2\) - \((-6x) (9) = -54x\)3. **Multiply Third Terms: (Distribute 11):** - \((11) (-8x^2) = -88x^2\) - \((11) (-7x) = -77x\) - \((11) (9) = 99\)Then, you sum all these results together: \[-16x^4 + 34x^3 - 28x^2 - 131x + 99\]This expanded polynomial is your product. Remember to combine like terms if necessary for simplification.