Question

Find the product. \((7-x)(x-1)\)

Short answer

The product of \((7-x)(x-1)\) is \(-x^2 + 8x - 7\).

Step-by-step solution

Distribute the first term of the first bracket

Multiply the first term of the first bracket with each term in the second bracket. This results in \(7*x - 7*1 = 7x - 7\).

Distribute the second term of the first bracket

Next, multiply the second term of the first bracket, which is '-x', with each term in the second bracket. Therefore, \(-x*x + x*1 = -x^2 + x\). Pay attention to the fact that a negative number multiplied by a positive number gives a negative number and negative multiplied by negative gives a positive number.

Combine the results

Finally, add the results of step 1 and step 2 to get the final answer. So, \(7x - 7 - x^2 + x = - x^2 + 8x - 7\). Note that for a standard form of a quadratic equation, the term with x^2 is written first.

Key concepts

Distributive Property

Understanding the distributive property is essential when multiplying polynomials, such as in the exercise e\[ (7-x)(x-1) \].
The distributive property, sometimes known as distribution, allows us to multiply a single term across terms within a parenthesis, effectively 'distributing' the multiplication. When we apply this property, we take a term outside the parentheses and multiply it by each term inside the parentheses separately.
For example, let's consider the distribution of the term e\[7\] outside the bracket across the terms e\[x\] and e\[-1\] inside the bracket. This results in e\[7 \times x\] and e\[7 \times (-1)\], which simplifies to e\[7x - 7\].

The Significance of Signs

It's crucial to pay attention to the signs of the numbers when utilizing the distributive property. The multiplication of a negative and a positive number results in a negative number, while multiplying two negatives result in a positive. This attention to detail is necessary to avoid common errors and reach the correct solution.

Quadratic Equations

In solving the multiplication problem from the exercise, we encounter a fundamental form of algebraic expression known as a quadratic equation.
The standardized form of a quadratic equation is e\[ ax^2 + bx + c = 0 \],
where e\[a\], e\[b\], and e\[c\] are constants, and e\[a\] is not equal to zero. The expression e\[-x^2 + 8x - 7\] from our solution is indeed a quadratic equation. It features a squared term, a linear term, and a constant term, collectively representing a parabola when graphed.

Standard Form and Its Components

In e\[-x^2 + 8x - 7\], the term e\[-x^2\] signifies that the parabola opens downwards since the coefficient is negative. The e\[8x\] is the linear term that indicates the slope and the e\[-7\] is the constant term impacting the y-intercept of the graph. Understanding each part's role helps in graphing and solving quadratic equations.

Algebraic Expressions

Algebraic expressions like the one we obtained in the original exercise are combinations of numbers, variables (like e\[x\]), and arithmetic operations (addition, subtraction, multiplication, and sometimes division). They don't contain an equals sign as equations do.
In our exercise, e\[-x^2 + 8x - 7\] is an algebraic expression which is the result of the polynomial multiplication.

Variables and Coefficients

Notice how the expression has a variable e\[x\] with numbers in front of it (e\[-1\] and e\[8\]); these numbers are called coefficients. The coefficients give us crucial information about how the variable e\[x\] is scaled within the expression. The expression also has a constant term (e\[-7\]), which is a fixed number unaffected by the variable's value. Developing the skill to identify and work with these components is vital for anyone studying algebra.