Problem 36 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ (x-3)(2 x-1)(x-2) $$

Step-by-Step Solution

Answer

Question: Rewrite and simplify the given polynomial expression into standard form, find the coefficients, and determine the degree of the polynomial: (x-3)(2x-1)(x-2) Answer: The simplified polynomial in standard form is 2x^3 - 11x^2 + 17x - 6. The coefficients are a_0 = -6, a_1 = 17, a_2 = -11, and a_3 = 2. The degree of the polynomial is 3.

Step 1: Multiply the first two expressions

Start by multiplying the first two expressions: \((x-3)(2x-1)\). For that, let's apply the distributive law. $$(x-3)(2x-1) = x(2x-1) - 3(2x-1)$$

Step 2: Expand the expressions

Now, let's expand the expressions: $$x(2x-1) - 3(2x-1) = 2x^2 - x - 6x + 3$$

Step 3: Combine like terms and simplify

Combine like terms and simplify the expression obtained in Step 2: $$2x^2 - x - 6x + 3 = 2x^2 - 7x + 3$$

Step 4: Multiply the simplified expression with the remaining expression

Next, multiply the simplified expression obtained in Step 3 with the remaining expression \((x-2)\): $$(2x^2 - 7x + 3)(x-2)$$

Step 5: Apply the distributive law

Apply the distributive law to expand the expression: $$(2x^2 - 7x + 3)(x-2) = (2x^2 - 7x + 3)x - (2x^2 - 7x + 3)2$$

Step 6: Expand the expressions

Expand the expressions obtained in Step 5: $$(2x^3 - 7x^2 + 3x) - (4x^2 - 14x + 6)$$

Step 7: Combine like terms and simplify

Finally, combine like terms and simplify the expression: $$2x^3 - 7x^2 + 3x - 4x^2 + 14x - 6 = 2x^3 - 11x^2 + 17x - 6$$ Now, the polynomial is in its standard form: $$2x^3 - 11x^2 + 17x - 6$$ The coefficients are: \(a_0 = -6\) \(a_1 = 17\) \(a_2 = -11\) \(a_3 = 2\)

Step 8: Find the degree of the polynomial

The degree of the polynomial is the highest power of \(x\) that appears in the expression. In this case, it is \(3\). Therefore, the degree of the polynomial is 3.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two polynomials to form a new polynomial. This requires distributing each term in the first polynomial by each term in the second polynomial, and then combining the results. When multiplying two binomials, for instance, you apply each part of one binomial to the other, ensuring each term interacts with every term from the other polynomial.

In this exercise, we've started with multiplying i(x-3)(2x-1)(x-2) i. First, we multiplied (i(x-3)(2x-1) ), resulting in an expression. Then multiplied the resulting product with (i x-2) i. By applying associative property in polynomial multiplication, which allows performing operations in different pairs, calculations become manageable.

Distributive Law

The distributive law is a fundamental principle in algebra that allows us to multiply a sum by another number or expression. It is expressed as (a(b+c) = ab + ac) . This property is useful in polynomial multiplication since it helps simplify the multiplication of expressions.

During the multiplication process, we distribute each term of one polynomial into each term of the other. For example, when we multiply i(x-3) i by i(2x-1) i, each term of the first polynomial ( x, -3 ) is distributed throughout the second polynomial ( 2x, -1 ), resulting in new expressions which eventually combine into 2x^2 - x - 6x + 3 .

Recalling these rules can "untangle" seemingly complex multiplication processes.

Combine Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions and is fundamental when working with polynomials. Like terms are terms that have the same variable raised to the same power. By combining them, we condense the expression to make it simpler and more interpretable.

After we've distributed terms and expanded the expressions, we will find some terms are 'like' and can be combined. In this exercise, we see that after expanding 2x^2 - x - 6x + 3 becomes 2x^2 - 7x + 3 after merging -x and -6x into -7x . Recognizing and combining like terms eliminates redundancy and clarifies the expression's structure.

Degree of Polynomial

The degree of a polynomial is one of its defining features, indicating the highest power of the variable present in the polynomial. It tells us not only about the polynomial's behavior but also its complexity. To find the degree, identify the term with the largest exponent of the variable.

In our example, the polynomial 2x^3 - 11x^2 + 17x - 6 has a highest power of 3 in 2x^3 . Therefore, the degree of this polynomial is 3.

Degrees help in predicting the general shape of a graph and understanding potential intersections with axes, providing crucial insights into solving polynomial equations.