Answered

Which point is a point where the graph of $y=(x+2)\left(x^2+4 x+3\right)$ crosses the $x$-axis?


A$(-3,0)$
B$(2,0)$
C$(1,0)$
D$(3,0)$

Verified Answer :

  • Set the function \(y = (x+2)(x^2 + 4x + 3)\) to zero to find the x-intercepts.
  • Factor the quadratic \(x^2 + 4x + 3\) into \((x+1)(x+3)\).
  • Solve the equation \((x+2)(x+1)(x+3) = 0\) to find the roots \(x = -2, -1, -3\).
  • Match the roots to the provided options, identifying \((-3, 0)\) as the correct point.
  • Final answer: \(\boxed{(-3,0)}\).

Explanation

  1. Analysis of the problem

To find the points where the graph of \(y = (x+2)(x^2 + 4x + 3)\) crosses the x-axis, we need to find the values of \(x\) for which \(y=0\). This leads to the equation:
\[(x+2)(x^2 + 4x + 3) = 0\]
This equation is satisfied if either \(x+2 = 0\) or \(x^2 + 4x + 3 = 0\).

  1. Factoring the equation

First, consider the term \(x+2 = 0\), which gives the root \(x = -2\). Next, we factor the quadratic expression \(x^2 + 4x + 3\). We look for two numbers that multiply to \(3\) and add up to \(4\). These numbers are \(1\) and \(3\). Thus, the quadratic can be factored as:
\[x^2 + 4x + 3 = (x+1)(x+3)\]
Substituting this back into the equation, we have:
\[(x+2)(x+1)(x+3) = 0\]
Applying the zero-product property, the graph crosses the x-axis at \(x = -2\), \(x = -1\), and \(x = -3\).

  1. Evaluation of the options

We are given the points \((-3, 0), (2, 0), (1, 0),\) and \((3, 0)\). We compare the x-coordinates of these points with the roots found in the previous step: \(\{-3, -2, -1\}\).
Comparing the options:

  • Point \((-3, 0)\) has \(x = -3\), which is a root.
  • Point \((2, 0)\) has \(x = 2\), which is not a root.
  • Point \((1, 0)\) has \(x = 1\), which is not a root.
  • Point \((3, 0)\) has \(x = 3\), which is not a root.

Therefore, the only point where the graph crosses the x-axis among the given options is \((-3, 0)\).

Examples

In real-life scenarios, finding the points where a function crosses the axis (the zeros or roots) is essential in optimization problems. For instance, if you are modeling the trajectory of a ball, the x-intercepts can represent the points where the ball hits the ground. Understanding how to factor these equations helps in predicting these critical events, allowing for better planning in engineering, sports science, and economics.

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